an optimal execution problem with s … optimal execution problem with s-shaped market impact...

AN OPTIMAL EXECUTION PROBLEM WITH S-SHAPED

MARKET IMPACT FUNCTIONS

TAKASHI KATO

Abstract. In this study, we extend the optimal execution problem with con-vex market impact function studied in Kato [14] to the case where the market

impact function is S-shaped, that is, concave on [0, x0] and convex on [x0,∞)for some x0 ≥ 0. We study the corresponding Hamilton–Jacobi–Bellmanequation and show that the optimal execution speed under the S-shaped

market impact is equal to zero or larger than x0. Moreover, we provide someexamples of the Black–Scholes model. We show that the optimal strategyfor a risk-neutral trader with small shares is the time-weighted average pricestrategy whenever the market impact function is S-shaped.

1. Introduction

Optimal execution problems have been widely investigated in mathematicalfinance as a type of stochastic control problem. There are various studies ofoptimal execution, such as [1, 2, 4, 7, 23] and references therein, and Gatheraland Schied [8] survey several dynamic models of optimal execution. To study thistype of problem, we cannot ignore market impact (MI), which is a market liquidityproblem. Here, we consider a situation where a single trader has many shares ofa security and tries to sell (liquidate) it until a time horizon. A large selling orderinduces a gap between supply and demand, causing a decrease in the securityprice. This effect is called the MI, and the trader should reduce the liquidationspeed to avoid the MI cost. However, reducing the liquidation speed also increasesthe timing cost, which is caused by the random fluctuation of the security priceover time. The trader should optimize the execution strategy by considering theMI cost and the timing cost. Therefore, the MI function, g(x), plays an importantrole in studying optimal execution problems. Here, g(x) implies the decrease ofthe security price by selling x shares (or selling rate).

The simplest setting for g is a linear function. For instance, in [2, 4, 23], optimalexecution problems are treated mainly with linear MI functions and derive optimalexecution strategies. However, there are studies on optimization problems withnon-linear g [1, 10, 11, 12, 14, 15]. In particular, we derive a mathematicallyadequate continuous-time model of an optimal execution problem as a limit ofdiscrete-time optimization problems in [14] when g is strictly convex.

Received 2017-6-29; Communicated by Hui-Hsiung Kuo.2010 Mathematics Subject Classification. Primary 91G80; Secondary 93E20, 49L25.Key words and phrases. Optimal execution problem, market impact, the Hamilton–Jacobi–

Bellman equation, time-weighted average price (TWAP) strategy.

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Serials Publications www.serialspublications.com

Communications on Stochastic Analysis Vol. 11, No. 3 (2017) 265-285

266 TAKASHI KATO

It is still unclear what form of g is natural. Recently, it has been proposedthat an S-shaped function is suitable for g. That is, g(x) should be concave on[0, x0] and convex on [x0,∞) for some x0 ≥ 0. Many traders intuitively expectthat MI functions are S-shaped [15]. Moreover, in [24], we find an empiricalprediction for a hump-shaped limit order book, which corresponds to the S-shapedMI function. Therefore, our previous study [14] should be extended to include theS-shaped MI function, g. We have tackled this problem partially in [15], but thereare still many mathematical and financial questions at this stage. For instance,when we consider the optimal execution problem with S-shaped g, it is intuitivethat the optimal execution speed should not reach the range (0, x0]. However,we have not proved this finding mathematically. Moreover, we have not discussedthe Hamilton–Jacobi–Bellman (HJB) equations corresponding to our optimizationproblem sufficiently.

In this paper, we resolve these questions as a continuation of our previous study[15]. We completely generalize our previous results [14] to the case of S-shapedg and study the above questions. Moreover, we find that the optimal executionstrategy of a risk-neutral trader in the Black–Scholes market model is the time-weighted average price (TWAP) strategy, that is, to sell at a constant speed.This is the same result as Theorem 5.4(ii) in [14] when g is a quadratic function;however, it is not necessary to assume an explicit form of g. We show that thisresult is true whenever g is S-shaped. This result generalizes Theorem 5.4(ii) in[14] and provides an analytical solution to the optimal execution problem with anuncertain MI given in Section 5.2 of [12].

The rest of this paper is as follows. In Section 2, we introduce a mathematicalmodel of an optimization problem based on our previous work [15] and review theprevious results. In Section 3, we characterize our value function as a viscositysolution to the corresponding HJB equation. We show the uniqueness of the vis-cosity solutions to the HJB equation under adequate conditions. To investigatethe properties of optimal strategies, we introduce a verification theorem and showthat the optimal execution strategy does not take the value in (0, x0] in Section4. In Section 5, we present some examples. In particular, we demonstrate the ro-bustness of the TWAP strategy as an optimal strategy in the Black–Scholes modelwith general shaped MI functions. We summarize our argument and introduce fu-ture tasks in Section 6. Section A gives supplemental arguments to guaranteeconsistency between our present model and our previous model [15]. All proofsare in Section B.

2. Model Settings

Let (Ω,F , (Ft)0≤t≤T , P ) be a stochastic basis and let (Bt)0≤t≤T be a one-dimensional Brownian motion (T > 0). Set D = R × [0,∞)2 and denote by Cthe set of non-decreasing, non-negative, and continuous functions with polynomialgrowth defined on D. For u ∈ C, we define a function, J(· ;u) : [0, T ]×D −→ R,as

J(t, c, x, s;u) = sup(xr)r∈At(x)

E[u(Ct, Xt, St)], (2.1)

OPTIMAL EXECUTION WITH S-SHAPED MARKET IMPACT 267

where (Cr)r, (Xr)r, and (Sr)r are stochastic processes given by

dCr = xrSrdr,

dXr = −xrdr,

dSr = b(Sr)dr + σ(Sr)dBr − Srg(xr)dr, (2.2)

and (C0, X0, S0) = (c, x, s), andAt(x) is the set of non-negative (Fr)r-progressively

measurable process, (xr)0≤r≤t, satisfying∫ t

0xrdr ≤ x a.s. We call an element of

At(x) an admissible strategy. Here, b and σ are defined as

b(s) =

(b(log s) +

1

2σ(log s)2

)s, σ(s) = σ(log s)s, s > 0

and b(0) = σ(0) = 0, where b, σ : R −→ R are bounded and Lipschitz continuousfunctions. g ∈ C([0,∞)) ∩ C1((0,∞)) is a non-negative function with g(0) = 0.

Function J implies the value function of an optimal execution problem with MIfunction g and is derived as a limit of discrete-time value functions in [14] wheng is convex, and in [15] when g is S-shaped, that is, when h := g′ satisfies thefollowing conditions.

[A1] h(x) ≥ 0, x > 0.[A2] limx→0 xh(x) = 0.[A3] There is an x0 ≥ 0 such that h is strictly decreasing on (0, x0] and strictly

increasing on [x0,∞).[A4] h(∞) = limx→∞ h(x) = ∞.

Condition [A3] implies that g is concave on [0, x0] and convex on [x0,∞). In thispaper, we always assume [A1]–[A4].

We briefly introduce the financial implications of our model (see [11, 12, 14]for more details). We assume that there is a single trader who has many sharesx0 of a security whose price is s0 at the initial time. The trader tries to sell thesecurity in the market until time horizon T , but the selling behavior affects thesecurity price via the effect of MI (denoted as the term −g(xr)dr in (2.3)). Sr isthe security price at time r, and Cr (respectively, Xr) describes the cash amount( respectively, shares of the security) held at time r. The trader’s purpose is tomaximize the terminal expected utility, J(T, c0, x0, s0;u) = E[u(CT , XT , ST )], bycontrolling an execution strategy, (xr)r ∈ AT (x). Here, xr implies the liquidationspeed at time r; in other words, the trader sells xrdr amount in the infinitesimaltime interval [r, r + dr]. To solve this problem, we introduce the value functionJ(t, c, x, s;u) for each t, c, x, and s to apply the dynamic programming method.

Remark 2.1.

(i) The log-price process Yr = logSr satisfies the stochastic differential equa-tion (SDE),

dYr = b(Yr)dr + σ(Yr)dBr − g(xr)dr, (2.3)

whenever Sr > 0, r ≥ 0.(ii) In stochastic control theory, (xr)r ∈ At(x) is called a control process

and (Cr, Xr, Sr)r defined in (2.2) is its controlled process. However, theexistence and uniqueness of (Cr, Xr, Sr)r for each (xr)r is not obvious in

268 TAKASHI KATO

our case, because (Yr)r may diverge due to the term −g(xr)dr. We canovercome this difficulty by regarding Sr = 0 after Yr diverges to −∞ (seeSection A for details).

(iii) In [11, 14, 15], we require an additional assumption such that each admis-sible strategy is essentially bounded; that is, we consider the optimizationproblem

J∞(t, c, x, s;u) = sup(xr)r∈A∞

t (x)E[u(Ct, Xt, St)] (2.4)

instead of (2.1), where

A∞t (x) =

(xr)r ∈ At(x) ; esssup

r,ωxr(ω) < ∞

.

This condition arises in the process of taking the limit from the discrete-time model to the continuous-time model; however, it is a mathematicaltechnical condition and is unnatural in relation to finance. We can showthat J coincides with J∞, and thus we are not overly concerned aboutthis problem (also see Section A).

In [15], we show that J(· ;u) is continuous on [0, T ]×D and J(r, · ;u) ∈ C foreach r ≥ 0 and u ∈ C. Moreover, J satisfies the dynamic programming principle,

J(t+ r, c, x, s;u) = J(t, c, x, s; J(r, ·;u)),

for each (c, x, s) ∈ D, u ∈ C and t, r ≥ 0 with t+ r ≤ T .By using these results, we characterize J as a viscosity solution of the corre-

sponding HJB equation in the next section. From now on, we fix u ∈ C and denoteJ(t, c, x, s;u) = J(t, c, x, s) for brevity.

3. Main Results I: Viscosity Properties

Our first main result is as follows.

Theorem 3.1.

(i) We assume that

lim infε→0

1

ε(J(t, c, x, s+ ε)− J(t, c, x, s)) > 0, (t, c, x, s) ∈ (0, T ]× D, (3.1)

where D = intD = R × (0,∞)2. Then J is a viscosity solution of the

following HJB equation on (0, T ]× D:

∂

∂tJ − sup

y≥0L yJ = 0, (3.2)

where

L y = (b(s)− sg(y))∂

∂s+

1

2σ(s)2

∂2

∂s2+ y

(s∂

∂c− ∂

∂x

).

(ii) We assume that equation (3.1) holds, b and σ are Lipschitz continuous, andlim infx→∞ h(x)/x > 0. Then we see the uniqueness of a viscosity solutionof (3.2) in the following sense. If a continuous function v : [0, T ]×D −→ R


with polynomial growth is a viscosity solution of (3.2) and satisfies theboundary conditions,

v(0, c, x, s) = u(c, x, s), (3.3)

v(t, c, 0, s) = E[u(c, 0, Zt(s))], (3.4)

v(t, c, x, 0) = u(c, x, 0), (3.5)

then it holds that J = v, where (Zr(s))r is a unique solution to the SDE:

dZr(s) = b(Zr(s))dr + σ(Zr(s))dBr, Z0(s) = s. (3.6)

Remark 3.2.

(i) The assertions of Theorem 3.1 are the same as those of Theorems 3.3 and3.6 in [14]. Thus, Theorem 3.1 was already obtained when g is convex,namely, when x0 = 0. As mentioned in Remark 3.7 of [14], our HJBequation (3.2) does not satisfy standard assumptions to apply a standardargument to viscosity characterization discussed in, for instance, [5, 6, 18,21]. We demonstrate Theorem 3.1(i) through a refinement of the proof ofTheorem 3.3 in [14]. In the proof of Theorem 3.6 in [14], we do not usethe convexity of g mainly, so Theorem 3.1(ii) is obtained in a similar wayto the proof of Propositions B.21–B.23 in [14].

(ii) The following condition is a standard natural condition for a utility func-tion in mathematical finance:[B] u(c, x, s) = U(c) for some concave function U ∈ C1(R).Under [B], the boundary conditions (3.3)–(3.5) are simplified as

v(0, c, x, s) = v(t, c, 0, s) = v(t, c, x, 0) = U(c).

(iii) It is not easy to check (3.1) in general. When g is convex, the natural andsimple sufficient conditions of (3.1) are introduced in [14] as[C1] u satisfies [B]. Moreover, it holds that U ′(c) ≥ δ, c ∈ R, for some

δ > 0.[C2] b and σ are differentiable and their derivatives are Lipschitz continu-

ous and uniformly bounded.In our case, by the same proof as for Proposition 3.5 in [14], we also verifythat (3.1) holds under [C1]–[C2].

4. Main Results II: Verification Arguments

Theorem 7.4 in [15] gives us a typical example where an optimal executionstrategy takes the value zero or larger than x0. This result is consistent withfinancial intuition, such as selling with the speed in the range of the concavepart of g (i.e., (0, x0]) induces superfluous transaction cost. In this section, wepresent a verification theorem to demonstrate that the optimal execution speed isin 0 ∪ (x0,∞) in general.

First, we introduce notation to state our second main result. Conditions [A3]and [A4] show that there is an inverse function, h−1 : [h(x0),∞) −→ [x0,∞), of

270 TAKASHI KATO

h. Then we define

H(s, p) =spc − px

sps1sps>0, (4.1)

Ξ(s, p) = h−1(H(s, p))1Λ(s, p) (4.2)

for s ≥ 0 and p = (pc, px, ps)′ ∈ R3, where

Λ =(s, p) ∈ (0,∞)× R3 ; ps > 0,H(s, p) > h(x0),

g(h−1(H(s, p))) < H(s, p)h−1(H(s, p)),

and A′ denotes the transpose of A. Moreover, for each continuously differentiablefunction, v : [0, T ]×D −→ R, we define

bv(t, c, x, s) =

sΞ(s,Dv(T − t, c, x, s))−Ξ(s,Dv(T − t, c, x, s))

b(s)− sg(Ξ(s,Dv(T − t, c, x, s)))

,

where D =(

∂∂c ,

∂∂x ,

∂∂s

)′.

Now, we present our second main result.

Theorem 4.1. We assume (3.1) and [B], that J ∈ C1,1,1,2((0, T ]×D), and that for

given (c0, x0, s0) ∈ D, there is a continuous process, (Ct, Xt, St)t, which satisfies

d

Ct

Xt

St

= bJ(t, Ct, Xt, St)dt+

00

σ(St)

dBt, t ∈ [0, τ ] (4.3)

and (C0, X0, S0) = (c0, x0, s0), where

τ = inft ≥ 0 ; (Ct, Xt, St) ∈ ∂D ∧ T.

Then there is an optimizer, (xt)t to J(T, c0, x0, s0), such that xt ∈ 0 ∪ (x0,∞),t ∈ [0, T ] a.s.

When executing a large amount of the security, it is important to decrease theexecution speed to reduce the execution cost. However, Theorem 4.1 tells us thatwhen the MI function is S-shaped (especially, concave on [0, x0]), it is undesirableto decrease the execution speed beyond the threshold, x0. An optimal executionstrategy in this case is to sell with the execution speed greater than x0 or to stopselling.

We can apply Theorem 4.1 if we verify the smoothness of the value functionJ . Even if we find a classical (sub)solution of (3.2), which does not necessarilysatisfy the boundary conditions (3.3)–(3.5), we can construct an optimal strategyto J(T, c0, x0, s0). We introduce the following verification theorem.

Theorem 4.2. Let (c0, x0, s0) ∈ D and let v ∈ C([0, T ]×D)∩C1,1,1,2((0, T ]× D)be a function that satisfies the following conditions.

(i) There are K,m > 0 such that

|v(t, c, x, s)| ≤ K(1 + cm + xm + sm), t ∈ [0, T ], (c, x, s) ∈ D.

(ii) v(0, c, x, s) ≥ u(c, x, s) holds for each (c, x, s) ∈ D.

(iii) ∂∂tv − supy≥0 L yv ≥ 0 on (0, T ]× D.


(iv) There is an (xt)t ∈ AT (x0) such that E[u(CT , XT , ST )] ≥ v(T, c0, x0, s0),

where (Ct, Xt, St)t is given by (2.2) with (C0, X0, S0) = (c0, x0, s0).

Then we have J(T, c0, x0, s0) = E[u(CT , XT , ST )] = v(T, c0, x0, s0) and (xt)t is itsoptimizer.

In the next section, we introduce some examples in which we derive optimalexecution strategies by using Theorem 4.2.

5. Examples

Similar to Section 5 in [14], we introduce some examples where the securityprice process is given as the Black–Scholes model. We assume that b(·) ≡ µ andσ(·) ≡ σ are constants and the utility function is set as uRN(c, x, s) = c; that is,the trader is risk-neutral.

By using the same argument as the proof of Proposition 5.2 in [14], we havethe following theorem.

Theorem 5.1. We have J(t, c, x, s) = c+ sW (t, x), where

W (t, x) = sup(xr)r∈Astat

t (x)

∫ t

0

exp

(−µr −

∫ r

0

g(xv)dv

)xrdr,

Astatt (x) = (xr)r ∈ At(x) ; (xr)r is deterministic,

µ = −µ− 1

2σ2.

Theorems 3.1 and 5.1 lead us to

Theorem 5.2. W is a viscosity solution to the partial differential equation

∂

∂tW + µW + inf

y≥0

Wg(y)−

(1− ∂

∂xW

)y

= 0 (5.1)

with the boundary condition

W (t, 0) = W (0, x) = 0. (5.2)

Moreover, if lim infx→∞ h(x)/x > 0, then a viscosity solution to (5.1)–(5.2) isunique in the following sense. If a continuous function w with polynomial growthis a viscosity solution to (5.1)–(5.2), then W = w.

Until the end of this section, we assume µ > 0 to focus on the case where theexpected security price decreases over time.

5.1. Mixed Power MI Function. Here, we consider the case where g is givenby

g(x) = βxπ (0 ≤ x ≤ x0), αxπ + γ (x > x0) (5.3)

for some x0 ≥ 0, α > 0 and 0 < π < 1 < π. Because g is continuously differen-tiable, β and γ must satisfy

β =π

παxπ−π

0 , γ =(ππ− 1)αxπ

0 .

Figure 1 shows the form of g when π = 0.5 and π = 2.The next result is a pure extension of Theorem 5.4 in [14].

272 TAKASHI KATO

Figure 1. Form of the MI function g(x) defined as (5.3) withπ = 0.5 and π = 2. The horizontal axis corresponds to x. Thevertical axis corresponds to g(x).

Theorem 5.3. Set

x∗,1 =1

δπB(1− exp

(− π

π − 1(µ+ γ)T

);1

π+ 1, 2

),

x∗,2 = νπT,

where

δπ = α1/ππ

(µ+ γ

π − 1

)π−1π

, νπ =

(µ+ γ

(π − 1)α

)1/π

and

B(z; a, b) =∫ z

0

dx

xa−1(1− x)b−1

is the incomplete Beta function.

(i) If x0 ≥ x∗,1, then we have

J(T, c0, x0, s0) = c0 +s0δπ

(1− exp

(− π

π − 1(µ+ γ)T

))π−1π

,

and its optimizer is given by

xt = νπ

(1− exp

(− π

π − 1(µ+ γ)(T − t)

))−1/π

.


(ii) If x0 ≤ x∗,2, then we have

J(T, c0, x0, s0) = c0 + s0 ·1− e−δπx0

δπ,


xt = νπ1[0,x0/νπ ](t).

Similarly to Theorem 5.4 in [14], the form of the optimal strategy changesdrastically according to the initial shares x0, and we do not have an analyticalsolution when x∗,2 < x0 < x∗,1. Moreover, when x0 ≤ x∗,2, the optimal strategyis the TWAP strategy, that is to sell with constant speed νπ. The TWAP strategyis the optimal strategy for the Almgren–Chriss model, which is a standard modelof optimal execution, for the risk-neutral trader [2, 7, 16, 17]. Theorem 5.3(ii) isalso obtained as a corollary of the result of the next subsection. In addition,

νπ >

(γ

(π − 1)α

)1/π

=

(π − π

π(π − 1)

)1/π

x0 > x0;

hence we can verify that xt ∈ 0 ∪ (x0,∞) in both cases of Theorem 5.3(i)(ii).This is consistent with Theorem 4.1.

5.2. TWAP Strategies for Small Amount Execution. Next, we considerthe case where the amount x0 of initial shares is small. Here, we do not restrictthe form of g without [A1]–[A4].

Before stating the result, we prepare the following proposition.

Proposition 5.4. Set Gh(x) = xh(x)−g(x). Then there is a unique νh ∈ (x0,∞)such that Gh(νh) = µ.

Theorem 5.5. If x0 ≤ νhT , then we have

J(T, c0, x0, s0) = c0 + s0 ·1− e−h(νh)x0

h(νh), (5.4)


xt = νh1[0,x0/νh](t). (5.5)

This theorem implies the robustness of the optimality of the TWAP strategyfor general shaped MI functions. When x0 is small, the optimal execution strategyis to sell the security at speed νh(> x0) until the time when the remaining sharesbecome zero.

Remark 5.6.

(i) As mentioned in Theorems 4.2 and 5.1 in [14], when g(x) = αx (α > 0) isgiven as a linear function, we have

J(T, c0, x0, s0) = c0 + s0 ·1− e−αx0

α, (5.6)

and the corresponding nearly optimal execution strategy is a quasi-blockliquidation with the initial time; that is, xδ

t = (x0/δ)1[0,δ](t) with δ → 0.This strategy formally corresponds to (5.5) taking the limit νh → ∞. Notethat h(x) ≡ α; hence (5.4) coincides with (5.6).

274 TAKASHI KATO

(ii) Let us consider an extreme case where

g(x) = g(x− x0)1[x0,∞)(x) (5.7)

for some increasing convex function g ∈ C1([0,∞); [0,∞)) with g(0) =g′(0) = 0 and g′(∞) = ∞. The form of g(x) is shown in Figure 2 for g(x)set as x3. In this case, we can completely avoid the MI cost by selling at aspeed lower than or equal to x0. Therefore, the optimal execution strategyseems to be xt = x01[0,x0/x0](t) at a glance. Following the strategy, (xt)t,we get the expected proceeds

C := s0

∫ x0/x0

0

e−µtx0dt = s0ι(µ/x0;x0),

where ι(y;x) = (1 − e−xy)/y. However, Theorem 5.5 implies that thisstrategy is not optimal; the optimal execution speed, νh, is strictly greaterthan x0. We compare the expected proceeds

C := J(T, 0, x0, s0) = s0ι(h(νh);x0)

obtained by the optimal strategy, (xt)t, with C obtained by (xt)t. Let us

denote G(x) = xg′(x) − g(x). Then we see that G(νh − x0) > G(0) = 0,and thus

µ = Gh(νh) = G(νh − x0) + x0h(νh) > x0h(νh).

This implies that h(νh) < µ/x0. Because ι(· ;x0) is decreasing, we have

C > C. This is because selling at a lower speed increases the executiontime and the timing cost. The trader should sell with the optimal speed,νh, and accept the MI cost.

5.3. Generalization of a Previous Result in Ishitani and Kato [12]. Asan application of Theorem 5.5, we provide an analytical solution to an optimalexecution problem with uncertain MI studied in Section 5.2 of [12]. We considerthe optimization problem

sup(xt)t∈AT (x0)

E

[∫ T

0

Stxtdt

], (5.8)

where (St)t is given by the SDE:

dSt = St(−µdt+ σdBt − g(xt)dLt), S0 = s0.

Here, (Lt)t is the Levy process, which is independent of (Bt)t and whose distribu-tion is given by the Gamma distribution

P (Lt − γt ∈ dz) =1

Γ(α1t)βα1t1

zα1t−1e−z/β11(0,∞)(z)dz,

where α1, β1, γ > 0 satisfy α1β1 ≤ 8γ and Γ(z) =∫∞0

tz−1e−tdt is the Gamma

function. Moreover, we assume that g(x) = α0x2 is given as a quadratic function

with α0 ≥ 0.In Section 5.2 of [12], we do not find the explicit form of the optimal strategy

to (5.8), even when x0 is small. However, numerical experiments suggest that the


Figure 2. Form of the MI function, g(x), defined as (5.7) withg(x) = x3. The horizontal axis corresponds to x. The verticalaxis corresponds to g(x).

optimal strategy with small x0 is the TWAP strategy. Here, we prove mathemat-ically that this conjecture is true.

Theorem 5.7. Let ν be the solution to

γα0ν2 + α1

2

(1− 1

1 + α0β1ν2

)− log(α0β1ν

2 + 1)

= µ.

If x0 ≤ νT , then the optimal strategy for (5.8) is given by the TWAP strategy

xt = ν1[0,x0/ν](t). (5.9)

6. Concluding Remarks

In this paper, we studied the optimal execution problem with S-shaped MI func-tions as a continuation of [15]. We showed that our value function is characterizedas a viscosity solution of the corresponding HJB equation. This is an extendedresult of that in [14]. Moreover, we provided the verification theorem to showthat the optimal execution speed is not in the range (0, x0]. This implies that thetrader should not blindly decrease the execution speed to reduce the MI cost.

In the Black–Scholes market model, we found that an optimal execution strategyis the TWAP strategy when the number of shares of the security held is small. Aconcrete form is not required for the MI function, g, so this result is robust andsuggests the optimality of TWAP strategy in practice.

276 TAKASHI KATO

The volume-weighted average price (VWAP) strategy is widely used in tradingpractice rather than the TWAP strategy [19]. Gatheral and Schied [8] pointed outthat we should regard the time parameter, t, not as physical time but as volumetime. Volume time implies a stochastic clock, which is measured by a markettrading volume process [3, 9, 20, 25]. If we consider the model on a volume timeline, we may find the optimality of the VWAP strategy in a similar way to Theorem5.5. However, we should not ignore the randomness of the market trading volume.One of our future tasks is to construct a model of optimal execution with S-shapedMI functions on a volume time line.

Furthermore, to apply Theorem 4.1, we require the value function, J , to besmooth, whereas it is difficult to show smoothness in general. Moreover, thesolvability of SDE (4.3) is not clear. Further study is needed.

Appendix A. Supplemental Arguments

We present the following propositions, which link the results in our previousstudy [15] with the present model.

Proposition A.1. Let t > 0 and let (c, x, s) ∈ D. For each (xr)r≤t ∈ At(x), thereis a unique process, (Cr, Xr, Sr)r≤t, that satisfies (2.2) and (C0, X0, S0) = (c, x, s).

The comparison theorem for solutions of SDEs (see Proposition 5.2.18 in [13]for instance) tells us that

0 ≤ Sr ≤ Zr(s) a.s., (A.1)

where (Zr(s))r is defined in (3.6). Moreover, Lemma B.1 in [14] tells us that

E[ sup0≤r≤t

Zr(s)m] < ∞ (A.2)

for each t, s and m > 0. Based on (A.1)–(A.2), we see that our value function,J(t, c, x, s), is well-defined and finite.

Proposition A.2. J(t, c, x, s) = J∞(t, c, x, s).

In [15], we show some properties of J∞(t, c, x, s). Proposition A.2 implies thatthese results also hold for J(t, c, x, s).

Appendix B. Proofs

Proof of Proposition 5.4. First, [A3] implies that

Gh(x0) = x0h(x0)−∫ x0

0

h(x)dx ≤ x0h(x0)− x0h(x0) = 0. (B.1)

[A3] also tells us that

Gh(x)−Gh(y) ≥ (h(x)− h(y))y > 0 (B.2)

for each x > y > x0. Hence, Gh is strictly increasing on (x0,∞). Moreover, lettingx → ∞ in (B.2), we see that

limx→∞

Gh(x) = ∞, (B.3)

owing to condition [A4]. Because Gh is continuous on [x0,∞) and µ is positive,(B.1)–(B.3) immediately give the assertion.


To show Proposition A.1, we prepare a lemma.

Lemma B.1. Let (φt)t be an (Ft)t-progressively measurable process such that|φt| ≤ K for some positive constant K. Then there is a CK,T > 0 that dependsonly on K and T , such that

E

[sup

0≤t≤Texp

(∫ t

0

φrdBr

)]≤ CK,T .

Proof. Put

Nt = exp

(∫ t

0

φrdBr −1

2

∫ t

0

φ2rdr

)− 1.

Ito’s formula immediately implies that (Nt)t is a continuous local martingale start-ing at 0 and d⟨N⟩t = (Nt + 1)2φ2

tdt.Take any R > 0 and define τR = inft ≥ 0 ; ⟨N⟩t ≥ R ∧ T and mR

t =

E[⟨N⟩t∧τR ](≤ R < ∞). Then we observe

0 ≤ mRt ≤ 2K2

E

[∫ t∧τR

0

(N2r + 1)dr

]≤ 2K2T + 2K2

∫ t

0

mRr dr.

We apply the Gronwall inequality to obtain

mRt ≤ 2K2T + 4K2T 2e2K

2T =: C ′K,T .

The Chebyshev inequality implies that τR T , R → ∞ a.s., and hence E[⟨N⟩T ] =limR→∞ mR

t ≤ C ′K,T by the monotone convergence theorem. Now we arrive at

E

[sup

0≤t≤Texp

(∫ t

0

φrdBr

)]≤ eK

2T/2(2E[⟨N⟩T ]1/2 + 1)

≤ eK2T/2

(2√

C ′K,T + 1

).

Proof of Proposition A.1. It suffices to show the existence and uniqueness of pro-cess (Sr)r≤t for each given (xr)r ∈ At(x) and s > 0.

Step 1. For each n ∈ N, define

τn = inf

r ≥ 0 ;

∫ r

0

g(xv)dv ≥ n

∧ t

and put xnr = xr1[0,τn](r). Then we can show that there is a unique solution (Y n

r )rto the following SDE by the standard argument:

dY nr = b(Y n

r )dr + σ(Y nr )dBr − g(xn

r )dr, Y n0 = log s.

Ito’s formula implies that the process Snr := exp(Y n

r ) satisfies

dSnr = b(Sn

r )dr + σ(Snr )dBr − Sn

r g(xnr )dr, Sn

0 = s.

We see that τn ≤ τm and Snr = Sm

r , r ∈ [0, τn] a.s. for each n < m. Therefore, wecan define S∞

r = limn→∞ Snr for each r ∈ [0, τ) ∩ [0, t] a.s., where τ = limn→∞ τn.

Next, we show that limr→τ S∞r = 0 a.s. on τ ≤ t. For each δ > 0, we see that

0 ≤ S∞τ−δ = lim

n→∞Snτ−δ ≤ sDtGδ on τ ≤ t,

278 TAKASHI KATO

where

Dt = lim infn→∞

sup0≤r≤t

exp

(∫ r

0

b(Y nv )dv +

∫ r

0

σ(Y nv )dBv

),

Gδ = exp

(−∫ τ−δ

0

g(xr)dr

).

Because b and σ are bounded, Lemma B.1 implies that E[Dt] < ∞, hence Dt < ∞a.s. Moreover, based on the definition of τ , it holds that Gδ1τ≤t −→ 0, δ → 0a.s. Thus, we have limδ→0 S

∞τ−δ = 0 a.s. on τ ≤ t.

Therefore, we can define Sr := S∞r∧τ as a continuous process on [0, t], and it

holds that

s+

∫ r

0

σ(Sv)dBv +

∫ r

0

(b(Sv)− Svg(xv))dv

= s+

∫ r∧τ

0

σ(S∞v )dBv +

∫ r∧τ

0

(b(S∞v )− S∞

v g(xv))dv

= limn→∞

Snr∧τn = Sr, r ≤ t.

Thus, (Sr)r satisfies (2.2).

Step 2. Next, we show the uniqueness of the solution to (2.2). Assume that (Sr)rsatisfies (2.2) and S0 = s. We see that Yr∧τn = logSr∧τn and Yr∧τn = log Sr∧τn

satisfy (2.3). Because b and σ are Lipschitz continuous, we have E[sup0≤r≤t |Yr∧τn

− Yr∧τn |2] = 0. This implies that Sr∧τn = Sr∧τn , r ≤ t a.s. Then we have

Sr∧τn = Sr∧τn , r ≤ t, a.s. Letting n → ∞, we arrive at Sr∧τ = Sr∧τ , r ≤ t

a.s. Based on (2.2), Sr = Sr = 0 for each r larger than τ a.s. on τ ≤ t, so we

conclude that (Sr)r is equal to (Sr)r a.s.

Proof of Proposition A.2. Because J(t, c, x, s) ≥ J∞(t, c, x, s) is clear, we mayprove the opposite inequality.

Fix any (xr)r ∈ At(x) and denote by (Cr, Xr, Sr)r≤t its controlled process.Take any K > 0 and set xK

r = xr ∧ K. Then (xKr )r ∈ A∞

t (x) holds. Let(CK

r , XKr , SK

r ) be the controlled process of (xKr )r. Then we have XK

t ≥ Xt.Moreover, Proposition 5.2.18 in [13] implies that SK

r ≥ Sr, r ≤ t a.s. Therefore,it holds that

CKt = c+

∫ t

0

xKr SK

r dr ≥ c+

∫ t

0

xKr Srdr a.s.,

and the monotone convergence theorem tells us that lim infK→∞ CKt ≥ Ct a.s.

Because u ∈ C, we have u(Ct, Xt, St) ≤ lim infK→∞ u(CKt , XK

t , SKt ). Then we

apply Fatou’s lemma to see that

E[u(Ct, Xt, St)] ≤ lim infK→∞

E[u(CKt , XK

t , SKt )] ≤ J∞(t, c, x, s).

Because (xr)r ∈ At(x) is arbitrary, we complete the proof.


To prove Theorem 3.1, we define F : D × R3 × S −→ R ∪ −∞ by

F (z, p,Σ) = −1

2σ(s)2Σss − b(s)ps +H(s, p),

H(s, p) = infy≥0

f(y; s, p),

f(y; s, p) = spsg(y)− (spc − px)y,

where S ⊂ R3 ⊗ R3 is the set of three-dimensional real symmetric matrices, andwe denote

z =

cxs

, p =

pcpxps

, Σ =

Σcc Σcx Σcs

Σxc Σxx Σxs

Σsc Σsx Σss

.

Note that (3.2) is equivalent to

∂

∂tJ + F (z,DJ,D2J) = 0. (B.4)

Moreover, put

U =(z, p,Σ) ∈ D × R3 × S ; F (z, p,Σ) > −∞

,

R = D × (R2 × (0,∞))× S .

Note that ps ≥ 0 holds for each (z, p,Σ) ∈ U . Moreover, we have R ⊂ U .

Lemma B.2. For each (z, p,Σ) ∈ R, we have

H(s, p) = f(h−1(H(s, p) ∨ h(x0)); s, p) ∧ 0 = f(Ξ(s, p); s, p), (B.5)

where H(s, p) and Ξ(s, p) are given by (4.1)–(4.2). In particular, F is continuouson R.

Proof. First, we note that H(s, p) = spsH(H(s, p)), where

H(y) = infy≥0

f(y; y), f(y; y) = g(y)− yy.

We see that

H(y) = f(h−1(y ∨ h(x0)); y) ∧ 0. (B.6)

Indeed, if y ≤ h(x0), we observe

∂

∂yf(y; y) = h(y)− y ≥ h(y)− h(x0) ≥ 0, y ≥ 0

by [A3]. Thus we get H(y) = f(0; y) = 0. Moreover, [A3] also implies

f(h−1(h(x0)); y) = f(x0; y) =

∫ x0

0

h(y′)dy′ − yx0 ≥ (h(x0)− y)x0 ≥ 0;

hence, (B.6) holds. In contrast, if y > h(x0), we see that f(· ; y) attains the mini-mum at h−1(y) or 0. When f(h−1(y); y) < 0, it holds that H(y) = f(h−1(y); y).When f(h−1(y); y) ≥ 0, it holds that H(y) = 0. In both cases, we see that (B.6)actually holds. (B.6) implies the first equality of (B.5). The second equality of(B.5) is obtained by a straightforward calculation using [A3]. The last assertion

is obtained by the continuity of b, σ, h, H, and H(s, p).

280 TAKASHI KATO

The following proposition is obtained by a standard argument (see [6, 18, 21]for details).

Proposition B.3. J is the viscosity supersolution of (3.2).

Proposition B.4. Assume (3.1). Then J is the viscosity subsolution of (3.2).

Proof. Fix each (t, z) ∈ (0, T ] × D. Let v ∈ C1,2((0, T ] × D) be a test function,such that J −v attains the local maximum, 0, at (t, z). Then we can find an r > 0such that

J(t′, z′) < v(t′, z′) (B.7)

holds for each (t′, z′) ∈ Br((t, z)) \ (t, z), where

Br((t, z)) = (t′, z′) ∈ (0, T ]× D ; |t′ − t|2 + |z′ − z|2 ≤ r2.

For each L > 0, define

FL(z, p,Σ) = −1

2σ(s)2Σss − b(s)ps + inf

0≤y≤Lf(y; s, p),

JL(t, c, x, s) = sup(xr)r∈AL

t (x)E[u(Ct, Xt, St)],

ALt (x) = (xr)r ∈ At(x) ; |xr| ≤ L.

Here, (Cr, Xr, Sr)r is given as (2.2) and (C0, X0, S0) = (c, x, s). Note that JL(t, c,x, s) J(t, c, x, s), L → ∞. By the same argument as Proposition B.18 in [14],we see that JL is a viscosity solution of

∂

∂tJL + FL(z,DJL,D2JL) = 0. (B.8)

Because JL − v is continuous, JL − v attains a maximum on the set Br((t, z));namely, there is a (tL, zL) ∈ Br((t, z)) such that maxBr((t,z))(J

L−v) = JL(tL, zL)−v(tL, zL). We show that

(tL, zL) −→ (t, z), L → ∞. (B.9)

Because (tL, zL)L is a bounded sequence, we see that for each increasingsequence, (Ln)n ⊂ (0,∞), there is a subsequence, (Lnk

)k, such that (tLnk, zLnk

)

converges to a point, (t∗, z∗) ∈ Br(t, z). Dini’s theorem implies that JL −→ J ,L → ∞ is uniform convergence on any compact set; thus, we see that

J(t∗, z∗)− v(t∗, z∗) = limk→∞

(JLnk (tLnk, zLnk

)− v((tLnk, zLnk

))) = 0.

Combining this with (B.7), we conclude that (t∗, z∗) must coincide with (t, z).Therefore, (B.9) is true.

Next, we define v ∈ C1,2((0, T ]× D) by

v(t′, z′) = v(t′, z′) + JL(tL, zL)− v(tL, zL).


Then we see that JL−v attains a local maximum, 0, at (tL, zL). Moreover, becauseJL is a viscosity solution to (B.8), it holds that

∂

∂tv(tL, zL) + FL(zL,D v(tL, zL),D

2v(tL, zL))

=∂

∂tv(tL, zL) + FL(zL,Dv(tL, zL),D

2v(tL, zL)) ≤ 0. (B.10)

Note that (zL,Dv(tL, zL),D2v(tL, zL)) ∈ R holds for large enough L. Indeed,(3.1) implies that (∂/∂s)v(t, z) > 0, and the convergence (tL, zL) −→ (t, z) andthe continuity of Dv lead us to (∂/∂s)v(tL, zL) > 0 for large enough L. Moreover,using Lemma B.2 and Dini’s theorem again, we see that FL converges to F asL → ∞ uniformly on any compact set in R. Therefore, taking L → ∞ in (B.10),we arrive at

∂

∂tv(t, z) + F (z,Dv(t, z),D2v(t, z)) ≤ 0.

This completes the proof.

Proof of Theorem 3.1. Assertion (i) is a consequence of Propositions B.3–B.4. As-sertion (ii) is obtained by the same arguments as the proofs of Propositions B.21–B.23 in [14]. We note that Proposition B.22 of [14] requires the convexity of g onlyon [x1,∞) for large enough x1.

We prepare the following lemma to show Theorem 4.1.

Lemma B.5. Assume [B] and that J ∈ C1,1,1,2((0, T ]×D). It holds that

∂

∂cJ(t, c, 0, s) = U ′(c), (B.11)

∂

∂xJ(t, c, 0, s) ≥ sU ′(c), (B.12)

∂

∂sJ(t, c, 0, s) =

∂

∂xJ(t, c, x, 0) = 0, (B.13)

∂

∂tJ(t, c, 0, s) =

∂

∂tJ(t, c, x, 0) = 0 (B.14)

for each t > 0 and (c, x, s) ∈ D.

Proof. (B.11), (B.13) and (B.14) are obtained from J(t, c, 0, s) = J(t, c, x, 0) =U(c).

To show (B.12), for each fixed t > 0 and each x ∈ (0, t2), set xr =√x1[0,

√x](r)

and let (Cr, Xr, Sr)r be the controlled process associated with (xr)r ∈ At(x).Then we see that

1

x(J(t, c, x, s)− J(t, c, 0, s))

≥ 1

xE[U(Ct)− U(c)] = E

[∫ 1

0

U ′(c+ kxAx)dkAx

], (B.15)

where Ax = 1√x

∫√x

0Srdr. By the Doob inequality, (3.18) in [11], and (A.1)–(A.2),

we have E[|Ax − s|] −→ 0, x → 0. In particular, Ax converges to s in probability.

282 TAKASHI KATO

Moreover, by (A.1)–(A.2) and the concavity of U , we have

E[ sup0≤x≤t2

A2x] < ∞, E[ sup

0≤k≤1,0≤x≤t2(U ′(c+ kxAx))

2] ≤ (U ′(c))2.

Therefore, we can apply the dominated convergence theorem to get

E

[∫ 1

0

U ′(c+ kxAx)dkAx

]−→ sU ′(c), x → 0. (B.16)

(B.15)–(B.16) lead us to (B.12).

Proof of Theorem 4.1. Define

xt = Ξ(St,DJ(T − t, Ct, Xt, St))1[0,τ)(t). (B.17)

Because XT∧τ ≥ 0, it holds that (xt)t ∈ AT (x0). Then Proposition A.1 implies

that there is a controlled process (Ct, Xt, St)t associated with (xt)t. We see that

Ct = Ct∧τ , Xt = Xt∧τ and St∧τ = St∧τ .Put τR = inft ≥ 0 ; St ≥ R ∧ (T − 1/R)+ for each R > 0. Note that (A.1)–

(A.2) and the Chebyshev inequality imply that τR T , R → ∞. Ito’s formulagives us

E[J(T − τR, CτR , XτR , SτR)]− J(T, c0, x0, s0)

= E

[∫ τR

0

(− ∂

∂tJ + L xtJ

)(T − t, Ct, Xt, St)dt

]. (B.18)

Based on Theorem 3.1(i), Lemma B.2, and the smoothness of J , we have(− ∂

∂tJ + L xtJ

)(T − t, Ct, Xt, St)

=

(− ∂

∂tJ + sup

y≥0L yJ

)(T − t, Ct, Xt, St) = 0, t < τ . (B.19)

On t ≥ τ, either Xt = 0 or St = 0 holds. If Xt = 0, we have(− ∂

∂tJ + sup

y≥0L yJ

)(T − t, Ct, 0, St)

= supy≥0

(StU

′(Ct)−∂

∂xJ(T − t, Ct, 0, St)

)y

= 0 =

(− ∂

∂tJ + L xtJ

)(T − t, Ct, 0, St), t ≥ τ (B.20)

by Lemma B.5. If St = 0, Lemma B.5 also implies that(− ∂

∂tJ + sup

y≥0L yJ

)(T − t, Ct, Xt, 0)

= 0 =

(− ∂

∂tJ + L xtJ

)(T − t, Ct, Xt, 0), t ≥ τ . (B.21)


Combining (B.18)–(B.21), we arrive at E[J(T − τR, CτR , XτR , SτR)] = J(T, c0, x0,

s0). Letting R → ∞, we have E[u(CT , XT , ST )] = J(T, c0, x0, s0) due to the dom-inated convergence theorem. Therefore, (xt)t is the optimizer to J(T, c0, x0, s0).Based on (4.2) and (B.17), we see that xt = 0 or xt > x0.

Proof of Theorem 4.2. Fix any (xt)t ∈ A∞T (x0) and n ∈ N, and let xn

t = (1 −1/n)xt. Denote by (Ct, Xt, St)t ( respectively, (C

nt , X

nt , S

nt )t) the controlled pro-

cess associated with (xt)t ( respectively, (xnt )t). Note that (A.1)–(A.2) imply

E[sup0≤t≤T (Snt )

m] < ∞ for each m > 0, and that (Cnt , X

nt , S

nt ) ∈ D, t ∈ [0, T ]

holds a.s.Take any R > 0 and set τR = inft ≥ 0 ; Sn

t ≥ R∧ (T −1/R)+. By a standardargument using Ito’s formula, we arrive at

E[v(T − τR, CnτR , X

nτR , S

nτR)]− v(T, c0, x0, s0)

≤ E

[∫ τR

0

(− ∂

∂tv + sup

y≥0L yv

)(T − t, Cn

t , Xnt , S

nt )dt

].

Combining this with assumption (iii), we have

E[v(T − τR, CnτR , X

nτR , S

nτR)] ≤ v(T, c0, x0, s0).

Here, based on assumptions (i)–(ii) and the Chebyshev inequality, we see thatτR T , R → ∞ a.s. and

E[u(CnT , X

nT , S

nT )] ≤ v(T, c0, x0, s0). (B.22)

Because (xt)t is essentially bounded, by using Theorem 2.5.9 in [18], we obtain

E[sup0≤t≤T | logSnt − logSt|4] −→ 0, n → ∞. Then we have E[sup0≤t≤T |Sn

t −St|2] −→ 0, and thus E[|Cn

T −CT |] −→ 0. Moreover, we see that E[|XnT −XT |] −→

0. Therefore, letting n → ∞ in (B.22) and applying Lemma B.2 in [14], we arriveat

E[u(CT , XT , ST )] ≤ v(T, c0, x0, s0).

Because (xt)t ∈ A∞T (x0) is arbitrary, we deduce that

J(T, c0, x0, s0) = J∞(T, c0, x0, s0) ≤ v(T, c0, x0, s0).

This and assumption (iv) lead to the conclusion that

J(T, c0, x0, s0) = E[u(CT , XT , ST )] = v(T, c0, x0, s0).

Theorems 5.3 and 5.5 are obtained by a straightforward calculation using The-orem 4.2.

Proof of Theorem 5.7. Theorem 5.2 in [12] tells us that (5.8) is equivalent withthe optimization problem

c0 + sup(xt)t∈AT (x0)

∫ T

0

Stxtdt,

284 TAKASHI KATO

where

dSt = −St(µ+ g(xt))dt, S0 = s0,

g(x) = γα0x2 + α1 log(α0β1x

2 + 1).

Moreover, we see that

g′′(x) ≥ α0α1β1(α0β1x2 − 3)2

4(α0β1x2 + 1)2,

hence, g(x) is strictly convex (see Corollary 1.3.10 in [22] for instance). Therefore,we can apply Theorem 5.5 to complete the proof. The optimal execution speed νsatisfies Gg′(ν) = µ, where

Gg′(x) = xg′(x)− g(x)

= γα0x2 + α1

2

(1− 1

1 + α0β1x2

)− log(α0β1x+ 1)

.

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18. Krylov, N. V.: Controlled Diffusion Processes, Springer, Berlin, 1980.

19. Madhavan, A.: VWAP Strategies, Trading 1 (2002) 32–39.20. Mazur, S.: Modeling market impact and timing risk in volume time, Algorithmic Finance 2

(2013), no. 2, 113–126.21. Nagai, H.: Stochastic Differential Equations, Kyoritsu Shuppan, Tokyo, 1999.

22. Niculescu, C. and Persson, L. -E.: Convex Functions and Their Applications, Springer, NewYork, 2006.

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Takashi Kato: Association of Mathematical Finance Laboratory (AMFiL), 2–10,

Kojimachi, Chiyoda, Tokyo 102-0083, JapanE-mail address: [email protected]

THE MOMENTS OF LÉVY’S AREA USING A STICKYSHUFFLE HOPF ALGEBRA

ROBIN HUDSON, UWE SCHAUZ, AND YUE WU

Abstract. Lévy’s stochastic area for planar Brownian motion is the dif-ference of two iterated integrals of second rank against its component one-dimensional Brownian motions. Such iterated integrals can be multipliedusing the sticky shuffle product determined by the underlying Itô algebra ofstochastic differentials. We use combinatorial enumerations that arise fromthe distributive law in the corresponding Hopf algebra structure to evaluatethe moments of Lévy’s area. These Lévy moments are well known to begiven essentially by the Euler numbers. This has recently been confirmedin a novel combinatorial approach by Levin and Wildon. Our combinatorialcalculations considerably simplify their approach.

1. Introduction

Lévy’s stochastic area is the signed area enclosed by the planar Brownian pathand its chord. It was originally defined rigorously by Lévy [13] and now is in-tensively applied in several areas of modern mathematics, such as rough pathanalysis.

Let B = (X, Y ) be a planar Brownian motion in terms of components X andY which are independent one-dimensional Brownian motions.

Definition 1.1. The Lévy area of B over the time interval [a, b) is the stochasticintegral

A[a,b) = 12

∫ b

a

((X − X(a)) dY − (Y − Y (a)) dX

).

In this definition the integral takes the same value whether it is regarded as ofItô or Stratonovich type, but in the remainder of this paper all stochastic integralswill be of Itô type, in contrast to [12] where the Stratonovich integral is used.

Lévy studied the characteristic function in [13, 14, 15, 16, 17]. He derived thefollowing formula:

Theorem 1.2. (Lévy [13])

E[exp

(izA[a,b)

)]= sech

( 12 (b − a) z

).

Received 2017-4-1; Communicated by the editors.2010 Mathematics Subject Classification. Primary 60J65; Secondary 05A15.Key words and phrases. Lévy area, sticky shuffle algebras, Euler numbers.

287



288 ROBIN HUDSON, UWE SCHAUZ, AND YUE WU

We can expand the right-hand side of the formula in Theorem 1.2 using theTaylor series

sech(z) =∞∑

m=0(−1)m A2m

(2m)!z2m , (1.1)

where the even Euler zigzag numbers A2m are related to the Riemann zeta functionζ by

ζ (2m) = π2m

(2m)!A2m . (1.2)

This expansion shows that the nonvanishing moments of the Lévy area A[a,b) aregiven by

E[A[a,b)

]2m =(

b − a

2

)2m

A2m . (1.3)

In [17], Lévy first showed Theorem 1.2 by using dyadic approximation. Hissecond proof is based on the skew product representation of planar Brownian mo-tion and depends on earlier work by Kac, Siegert, Cameron and Martin (see [17]).In 1980, Yor [21] simplified Lévy’s proof by employing a result on Bessel pro-cesses and an elementary result used by D.Williams. Shortly afterwards, Helmesand Schwane [6] revisited the problem by extending the 2-dimensional set-up con-sidered by Lévy to d ≥ 2 dimensions. They considered the joint characteristicfunction of the stochastic process paired by the d-dimensional Brownian motionW and a certain generalized Lévy area AJ,x

[0,t] given by

AJ,x[0,t] :=

∫ t

0[W (s) + x(s)]J(s)dW (s). (1.4)

Here, [W (s) + x(s)] is viewed as a row-vector, a 1 × d matrix. For d = 2, x = 0and J =

[ 0 1−1 0

]the process AJ,x

[0,t]/2 coincides with Lévy stochastic area A[0,t]in Definition 1.1. Recently Levin and Wildon in [12] used iterated integrals andcombinatorial arguments involving the shuffle product (see [7]) to prove Theorem1.2. Our method starts from the same iterated integrals. Hence, we use that theintegral in Definition 1.1 can be written as

A[a,b) = 12

∫a <x< y< b

(dX(x)dY (y) − dY (x)dX(y)

). (1.5)

We may thus evaluate the moments as expectations of powers, using the so-calledsticky shuffle [8] Hopf algebra. The multiplication in this algebra can be used toexpress the product of two iterated Itô stochastic integrals as a linear combinationof such iterated integrals. Since the expectation of an iterated integral vanishesunless each of the individual integrators is time, the recovery formula [1, 8] involv-ing higher order Hopf algebra coproducts reduces the evaluation of the momentsto a combinatorial counting problem.

We are anxious to understand the origin of the remarkable cancelations whichgive rise to the Euler numbers A2m in the moments. In this regard, our com-binatorial calculations are relatively direct. Whereas Levin and Wildon, in theirderivation [12], use Eulerian numbers, a refinement of Euler numbers, we do nothave to use these refinements in our calculations. Although, like in Levin and

THE MOMENTS OF LÉVY’S AREA 289

Wildon’s paper, the final result is not new in that everything can be derived fromLévy’s formula for the characteristic function, our general approach also opens thedoor to further generalizations. In a separate paper [11], we shall give a general-ization which cannot be derived with previous methods. In that paper, the planarBrownian motion is replaced by a one-parameter family of quantum or noncommu-tative deformations [3]. The moments of the Lévy areas of these quantum planarBrownian motions interpolate between the classical case of Levin and Wildon andthe already known Fock case [2], where they are all trivial.

The sticky shuffle Hopf algebra is reviewed in Section 2 and its use for reduc-ing the evaluation of moments to a counting problem is described in Section 3.In Section 4, we apply combinatorial tools to show the main result. Finally, inthe appendix, we provide a simple lemmas about Euler numbers needed in ourcalculations.

2. The Sticky Shuffle Product Hopf Algebra

Let there be given an associative algebra L over C. The corresponding vectorspace T (L) of tensors of all ranks over L is defined as

T (L) =∞⊕

n=0

n⊗j=1

L . (2.1)

We denote by (α0, α1, α2, ...) the general element α = α0 ⊕ α1 ⊕ α2 ⊕ · · · of T (L),where only finitely many of the αm are nonzero. For each αm ∈

⊗mj=1 L the

corresponding embedded element (0, 0, ..., αm, 0, ...) of T (L) is denoted by αm .In the following, we use the notational convention that, for arbitrary elements αof T (L) and L of L, α ⊗ L is the element of T (L) for which (α ⊗ L)0 = 0 and(α ⊗ L)n = αn−1 ⊗ L for n ≥ 1.

The so-called sticky shuffle product Hopf algebra over L is formed by equippingT (L) with the operations of product, unit, coproduct and counit defined as follows.

• The sticky shuffle product of arbitrary elements of T (L) is defined induc-tively by bilinear extension of the rules

1C L1 ⊗ L2 ⊗ · · · ⊗ Lm = L1 ⊗ L2 ⊗ · · · ⊗ Lm 1C= L1 ⊗ L2 ⊗ · · · ⊗ Lm , (2.2)

L1 ⊗ L2 ⊗ · · · ⊗ Lm Lm+1 ⊗ Lm+2 ⊗ · · · ⊗ Lm+n= (L1 ⊗ · · · ⊗ Lm−1 Lm+1 ⊗ · · · ⊗ Lm+n) ⊗ Lm

+ (L1 ⊗ · · · ⊗ Lm Lm+1 ⊗ · · · ⊗ Lm+n−1) ⊗ Lm+n (2.3)+ (L1 ⊗ · · · ⊗ Lm−1 Lm+1 ⊗ · · · ⊗ Lm+n−1) ⊗ LmLm+n.

• The unit element for this product is 1T (L) = (1C, 0, 0, ...).


• The coproduct ∆ is the map from T (L) to T (L) ⊗ T (L) defined by linearextension of the rules that ∆

(1T (L)

)= 1T (L) ⊗ 1T (L) = 1T (L)⊗T (L) and

∆ L1 ⊗ L2 ⊗ · · · ⊗ Lm= 1T (L) ⊗ L1 ⊗ L2 ⊗ · · · ⊗ Lm (2.4)

+m∑

j=2L1 ⊗ L2 ⊗ · · · ⊗ Lj−1 ⊗ Lj ⊗ Lj+1 ⊗ · · · ⊗ Lm

+ L1 ⊗ L2 ⊗ · · · ⊗ Lm ⊗ 1T (L).

• The counit ε is the map from T (L) to C defined by linear extension of

ε(1T (L)

)= 1C and ε L1 ⊗ L2 ⊗ · · · ⊗ Lm = 0 for m > 0. (2.5)

Remark 2.1. There is a useful alternative equivalent definition of the sticky shuffleproduct. We can define the product γ = αβ by

γN =∑

A∪B=1,2,...,N

αA|A|β

B|B|. (2.6)

Here the sum is now over the 3N not necessarily disjoint ordered pairs (A, B)whose union is 1, 2, ..., N, and the notation is as follows; |A| denotes the numberof elements in the set A so that α|A| denotes the homogeneous component of rank|A| of the tensor α = (α0, α1, α2, ...) , and αA

|A| indicates that this component isto be regarded as occupying only those |A| copies of L within

⊗Nj=1 L labelled by

elements of the subset A of 1, 2, ..., N. Thus with βB|B| defined analogously the

combination αA|A|β

B|B| is a well-defined element of

⊗Nj=1 L. Here, if A ∩ B = ∅,

double occupancy of a copy of L within⊗n

j=1 L is reduced to single occupancy byusing the multiplication in the algebra L as a map from L × L to L. That (2.6) isequivalent to (2.3) is seen by noting that the three terms on the right-hand side of(2.3) correspond to the three mutually exclusive and exhaustive possibilities thatN ∈ A ∩ Bc, N ∈ Ac ∩ B and N ∈ A ∩ B in (2.6).

The recovery formula [1] expresses the homogeneous components of an elementα of T (L) in terms of the iterated coproduct ∆(N)α by

αN =(

∆(N)α)

(1,1,...,(N)1 )

. (2.7)

Here, ∆(N) is defined recursively by

∆(2) = ∆ and ∆(N) =(∆ ⊗ Id⊗(N−2)(T (L))

) ∆(N−1) for N > 2 . (2.8)

Hence, it is a map from T (L) to the Nth tensor power⊗(N)T (L) =

⊗(N)

∞⊕n=0

n⊗j=1

L =∞⊕

n1,n2,...,nN =0

N⊗r=1

nr⊗jr=1

L (2.9)

so that ∆(N)α has multirank components α(n1,n2,...,nN ) of all orders. The recoveryformula (2.7) also holds when N = 0 and N = 1 if we define ∆(0) and ∆(1) to bethe counit ε and the identity map 1T (L) respectively.


Note that ∆ is multiplicative, ∆ (αβ) = ∆ (α) ∆ (β), where the product on thetensor square T (L) ⊗ T (L) is defined by linear extension of the rule

(a ⊗ a′)(b ⊗ b′) = ab ⊗ a′b′ . (2.10)

3. Moments and Sticky Shuffles

We now describe the connection between sticky shuffle products and iteratedstochastic integrals. We begin with the well-known fact that, for the one-dimen-sional Brownian motion X and for a ≤ b,

(X(b) − X(a))2 = 2∫

a≤x<b

(X(x) − X(a)) dX(x) +∫

a≤x<b

dT (x), (3.1)

where T (x) = x is time. We introduce the Itô algebra L =C ⟨dX, dT ⟩ of complexlinear combinations of the basic differentials dX and dT, which are multipliedaccording to the table

dX dTdXdT

dT 00 0

(3.2)

together with the corresponding sticky shuffle Hopf algebra T (L). For each pair ofreal numbers a < b, we introduce a map Jb

a from T (L) to complex-valued randomvariables on the probability space of the Brownian motion X by linear extensionof the rule that, for arbitrary dL1, dL2, · · · dLm ∈ dX, dT

Jba dL1 ⊗ dL2 ⊗ · · · ⊗ dLm

=∫

a≤x1<x2<···<xm<b

dL1(x1) dL2(x2) dL3(x3) · · · dLm(xm) (3.3)

=∫ b

a

· · ·∫ x4

a

∫ x3

a

∫ x2

a

dL1(x1) dL2(x2) dL3(x3) · · · dLm(xm).

By convention Jba maps the unit element of the algebra T (L) to the unit random

variable identically equal to 1.Then (3.1) can be restated as follows,

Jba (dX) Jb

a (dX) = Jba (dX dX) , (3.4)

using the fact that dX2 = 2 dX ⊗ dX + dT .The following more general Theorem is probably known to many probabilists.

Theorem 3.1. For arbitrary α and β in T (L),

Jba(α)Jb

a(β) = Jba(αβ) .

Proof. By bilinearity it is sufficient to consider the case when

α = dL1 ⊗ dL2 ⊗ · · · ⊗ dLm , β = dLm+1 ⊗ dLm+2 ⊗ · · · ⊗ dLm+n (3.5)

for dL1, dL2, · · · , dLm+n ∈ dX, dT . In this case Theorem 3.1 follows, using theinductive definition (2.3) for the sticky shuffle product, from the product form ofItô’s formula,

d (ξη) = (dξ) η + ξdη + (dξ) dη (3.6)


where stochastic differentials of the form dξ = FdX + GdT, with stochasticallyintegrable processes F and G, are multiplied using the table (3.2).

For planar Brownian motion B = (X, Y ) the Ito table (3.2) becomesdX dY dT

dXdYdT

dT 00 dT

00

0 0 0

(3.7)

Corollary 3.2. Theorem 3.1 holds when L is the algebra defined by the multipli-cation table (3.7).

Basic for our calculations is the next theorem. It follows from the fact thatexpectations of stochastic integrals against either dX or dY as integrators arezero.

Theorem 3.3. For arbitrary n ∈ N, a < b ∈ R and basis elements dL1, dL2, . . . ,dLn ∈ dX, dY, dT,

E[Jb

a dL1 ⊗ dL2 ⊗ ... ⊗ dLn]

= 0unless dL1 = dL2 = · · · = dLn = dT.

In view of (1.5)

A[a,b) = 12

Jba(dX ⊗ dY − dY ⊗ dX) . (3.8)

Now consider the moments sequence of classical Lévy area in terms of the basis(dX, dY, dT ), i.e., Eqn. (3.8). In view of Theorem 3.1[

A[a,b)]n = 1

2n

(Jb

a(dX ⊗ dY − dY ⊗ dX))n

= 12n

Jba (dX ⊗ dY − dY ⊗ dXn) . (3.9)

The nth sticky shuffle power dX ⊗ dY − dY ⊗ dXn will consist of non-stickyshuffle products of rank 2n together with terms of lower ranks n, n+1, ..., 2n−1, allof which except the rank n term will contain one or more copies of dX and dY ,and will thus not contribute to the expectation in view of Theorem 3.3. The term

of rank n will be a multiple of dT ⊗ dT · · · ⊗(n)dT . Thus we can write

dX ⊗ dY − dY ⊗ dXn = wn

dT ⊗ dT · · · ⊗

(n)dT

+ terms of rank > n. (3.10)for some coefficient wn. The corresponding moment is given by

E[A[a,b)

]n = wn

2nE

[Jb

a

(dT ⊗ dT · · · ⊗

(n)dT

)]

= wn

2n

∫a≤x1<x2<···<xn<b

dx1 dx2 · · · dxn

= wn (b − a)n

2nn!. (3.11)


By the recovery formula (2.7) and the multiplicativity of the nth order coprod-uct ∆(n),

wndT ⊗ dT · · · ⊗(n)dT

= dX ⊗ dY − dY ⊗ dXnn

=(

∆(n)(dX ⊗ dY − dY ⊗ dXn))(1,1,...,

(n)1 )

=((

∆(n)(dX ⊗ dY − dY ⊗ dX))n)

(1,1,...,(n)1 )

. (3.12)

Now

∆(n)(dX ⊗ dY − dY ⊗ dX)

=∑

1≤j≤n

1T (L) ⊗ · · · ⊗(j)

dX ⊗ dY − dY ⊗ dX ⊗ · · · ⊗ 1T (L)

+∑

1≤j <k≤n

(1T (L) ⊗ · · · ⊗

(j)dX ⊗ · · · ⊗

(k)dY ⊗ · · · ⊗

(n)1T (L)

− 1T (L) ⊗ · · · ⊗(j)

dY ⊗ · · · ⊗(k)

dX ⊗ · · · ⊗(n)

1T (L)

). (3.13)

The first term of this sum, being of rank 2, cannot contribute to the component of

joint rank (1, 1, ...,(n)1 ) of the nth power of ∆(n) (dX ⊗ dY − dY ⊗ dX), where

product in the nth tensor power⊗(N)T (L) is defined exactly as in the case n = 2

in (2.10). Thus

wndT ⊗ dT · · · ⊗(n)dT

=((

∆(n)(dX ⊗ dY − dY ⊗ dX))n )

(1,1,...,(n)1 )

(3.14)

=

∑1≤j <k≤n

(1T (L) ⊗ · · · ⊗

(j)dX ⊗ · · · ⊗

(k)dY ⊗ · · · ⊗

(n)1T (L)

− 1T (L) ⊗ · · · ⊗(j)

dY ⊗ · · · ⊗(k)

dX ⊗ · · · ⊗(n)

1T (L)

))n)(1,1,...,

(n)1 )

.

This calculation of wn can be finished using some combinatorics. We do that inthe following section.

4. The Moments of Lévy’s Area

To evaluate the moments E[A[a,b)

]n, we need to calculate the number wn, asexplained in (3.11). By (3.14), we have

wn dT ⊗ dT · · · ⊗(n)dT =

∑h =k

sn(h, k)Rh,k

n(1,1,...,

(n)1 )

(4.1)


with

Rh,k := 1 ⊗ · · · ⊗ 1 ⊗(h)

dX ⊗ 1 ⊗ · · · ⊗ 1 ⊗(k)

dY ⊗ 1 ⊗ · · · ⊗ 1 (4.2)

and

sn(h, k) :=

+1 if h < k,

−1 if h > k.(4.3)

The nth power in (4.1) is based on the sticky shuffle product in T (L) and itsextension to the nth tensor power

⊗(n)T (L), as described in (2.10) for n = 2.If we set e := (h, k), then we may also write Re for Rh,k and sn(e) for sn(h, k).

Using distributivity, this yields

wn dT ⊗ dT · · · ⊗(n)dT =

∑ (n∏

ℓ=1

sn(eℓ)

)(n∏

ℓ=1

Reℓ

)(1,1,...,

(n)1 )

, (4.4)

where the sum runs over all n-tuples (e1, e2, . . . , en) of pairs (h, k) with h = k.We may imagine each pair eℓ = (hℓ, kℓ) as a directed edge with label ℓ, an arc,from hℓ to kℓ. Each n-tuples (e1, e2, . . . , en) is then a directed labeled multigraph,we say a digraph, on the vertex set V := 1, 2, . . . , n. We have to see whatthe individual arcs eℓ of a digraph (e1, e2, . . . , en) contribute to its correspondingsummand ±

∏nℓ=1 Reℓ

inside the sum (4.4). For example, in the case n = 4, thetwo arcs e1 = (1, 2) and e2 = (3, 2) contribute

sn(e1) sn(e2)(Re1Re2

)(1,1,1,1)

= −((dX ⊗ dY ⊗ 1 ⊗ 1)(1 ⊗ dY ⊗ dX ⊗ 1)

)(1,1,1,1)

= −(dX1

)(1) ⊗

(dY dY

)(1) ⊗

(1dX

)(1) ⊗

(1 · 1

)(1)

= − dX ⊗ dT ⊗ dX ⊗ 1 ,

(4.5)

where we basically ignored e3 and e4 (and the corresponding Re3 , Re4 , sn(e3) andsn(e4)) to illustrate how the product operates.

In order to calculate the coefficient wn of dT ⊗dT ⊗· · ·⊗dT in (4.4), we need toretain only those summands ±

∏nℓ=1 Reℓ

that contribute a scalar multiple of dT ⊗dT ⊗· · ·⊗dT . We may discard other summands. Hence, we do not have to sum overall digraphs (e1, e2, . . . , en). To see which ones we have to retain, let us assume that(e1, e2, . . . , en) yields a multiple of dT ⊗dT ⊗· · ·⊗dT in (4.4). Since the n copies ofdX and n copies of dY in the unexpanded product

∏nℓ=1 Reℓ

must yield n copiesof dT, one in each possible position, each vertex of the digraph (e1, e2, . . . , en)must have either exactly two incoming and no outgoing arcs (corresponding to a(dY )2 ) or exactly two outgoing and no incoming arcs (corresponding to a (dX)2 ).This shows that, inevitable, (e1, e2, . . . , en) must consist of disjoint alternatinglyoriented cycles that cover V, cycles whose arcs go “forward - backward - forward- backward - . . . ”. Every such digraph (e1, e2, . . . , en) has necessarily an evennumber of vertices, n = 2m, and contributes either +1 or −1 to w2m. Hence,

w2m =∑ 2m∏

ℓ=1

sn(eℓ) , (4.6)


where the sum runs over all diagraphs (e1, e2, . . . , e2m) that consist of disjointalternatingly oriented cycles. For odd n we do not obtain any term dT ⊗ dT ⊗· · · ⊗ dT in (4.4), i.e. w2m+1 = 0.

To further simplify the purely combinatorial expression in (4.6), we transformthe alternatingly oriented digraphs (e1, e2, . . . , e2m) into cyclically oriented di-graphs D, and, eventually, into permutations of a certain kind. Turning aroundeach second arc in each cycle, we get cyclicly oriented cycles (like cyclic one wayroads). These disjoint cycles still cover V = 1, 2, . . . , 2m and have even length, asthey arose from alternatingly oriented cycles. By performing that transformation,we do not change the numbers, as we can always go back to alternatingly orientedcycles by flipping each second arc1. The change from alternatingly to cyclicallyoriented cycles only results in an additional factor of (−1)m in our calculations.Apart from this, formula (4.6) stays the same if we alter the summation range tothe set of all diagraphs (e1, e2, . . . , e2m) that consist of disjoint cyclically orientedcycles of even length. Moreover, at this point, our cyclically oriented digraphs donot contain multiple arcs, so that we may forget the labels 1, 2, . . . , 2m of the arcse1, e2, . . . , e2m. In other words, we may replace the 2m-tuples (e1, e2, . . . , e2m) inthe summation range with the corresponding sets e1, e2, . . . , e2m. If a digraphwith 2m many unlabeled arcs has no multiple (no indistinguishable) arcs, then itcorresponds to exactly (2m)! many labeled digraphs, yielding a factor of (2m)! inour sum2. Thus,

w2m = (−1)m(2m)!∑ 2m∏

ℓ=1

sn(eℓ) , (4.7)

where the sum is now running over all unlabeled diagraphs e1, e2, . . . , e2m thatconsist of disjoint cyclically oriented cycles of even length.

Eventually, we can now turn towards permutations in the symmetric groupS2m as representatives for diagraphs D = e1, e2, . . . , e2m. We may view eacharc (h, k) ∈ D in any cyclically oriented unlabeled digraph D as the assignmentof a function value, h 7→ k =: s(h), and obtain a permutation3 s = sD on V =1, 2, . . . , 2m. In our case, the cycles of s have even length. Denoting withD2m ⊆ S2m the set of all permutations whose cycles have even length, we see that

w2m = (−1)m(2m)!∑

s∈D2m

sn(s) , (4.8)

where

sn(s) :=2m∏j=1

sn(j, s(j)) . (4.9)

1To be precise, each (labeled) even cycle has two alternating orientations but also two cyclicorientations. So, if we allow only even cycles, then every system of cycles has as many cyclic asalternating orientations.

2Very careful readers might be astonished that we could not drop the edge labels earlier inthis way. We invite them to investigate the case m = 1 to see why.

3Every permutation s ∈ S2m is a bijective map from V to V. Since every map is a relation,this means that s is a subset of V × V. In this sense, sD := D.


Figure 1. The cyclic permutation s = (4 1 8 2 6 7 5 3) with small-est transit h = 5.

To determine this sum, we cancel off some summands s with opposite signssn(s). We call a point h ∈ V a transit of s if

either s−1(h) < h < s(h) or s−1(h) > h > s(h) . (4.10)

Let D′2m bet the set of all s ∈ D2m which have at least one transit. We show

that the elements of D′2m cancel out completely and can be ignored in our sum.

Obviously, every s ∈ D′2m contains a unique smallest transit h, and we obtain a

permutation s′ of V \h by replacing the chain of assignments s−1(h) 7→ h 7→ s(h)with the shorter chain s−1(h) 7−→ s(h). The new permutation s′ has a unique oddcycle

j1 7→ j2 7→ · · · 7→ s−1(h) 7−→ s(h) 7→ · · · 7→ j2k−1 7→ j1. (4.11)If we walk once around this cycle and observe the indices j1, j2, j3, . . . as a kindof altitude, then we will cross the altitude h as many times upwards, from belowh to above h, as downwards, from above h to below h. Hence, there is an evennumber of ways to reinsert h as transit into that odd cycle, see Fig. 4. One halfof the permutations that we obtain will have positive sign, one half negative sign.Removal and reinsertion of a smallest transit yields an equivalence relation ∼on D′

2m. We have s ∼ r if and only if s′ = r′. The corresponding equivalenceclasses form a partition of D′

2m, and each of them cancels off nicely. For example,in Fig4, the four permutations, (4 1 8 2 6 7 5 3), (4 1 8 2 5 6 7 3), (4 1 8 5 2 6 7 3) and(4 1 5 8 2 6 7 3) are the only permutations yielding the odd cycle (4 1 8 2 6 7 3), if thesmallest transit is removed. They form one equivalence class. And, as the readermay check, its elements actually cancel out, as two of them have positive sign andtwo have negative sign. So, indeed, we only have to sum over D2m\D′

2m, that is,over all permutations s ∈ S2m with

either s−1(j) < j > s(j) or s−1(j) > j < s(j) (4.12)

for all j ∈ V := 1, 2, . . . , 2m. We call this kind of permutations forth-back permu-tations. Their number is the so-called Euler zigzag number A2m, i.e. |D2m\D′

2m| =


A2m, as shown in Lemma 5.1 in the appendix. Since all forth-back permutationshave sign (−1)m, we get

w2m = (2m)! A2m. (4.13)From this and Equation (3.11), we finally arrive at Lévy’s classical result (1.3):

Theorem 4.1. The nonzero moments of the Lévy area A[a,b) are

E[A[a,b)

]2m =(

b − a

2

)2m

A2m .

5. Appendix About Euler Numbers

In this section, we present a simple lemmas about Euler numbers. It is ofsufficient general nature to be of potential interest elsewhere. Many similar resultsand basics can be found in [19] and [20].

A permutation s in the symmetric group Sn is a zigzag permutation (mislead-ingly also called alternating permutation) if s(1) > s(2) < s(3) > s(4) < · · · . Inother words, s is zigzag if s(1) > s(2) and

either s(j − 1) < s(j) > s(j + 1) or s(j − 1) > s(j) < s(j + 1) (5.1)for all j ∈ 2, 3 . . . , n − 1. If we have the initial condition s(1) < s(2), instead ofs(1) > s(2), we may call s zagzig. The number of all zigzag permutations in Sn

is called the Euler zigzag number An. These numbers occur in many places, forexample, as the coefficients of z2n

(2n)! in the Maclaurin series of sec(z) + tan(z). Inthis paper, we met them as the number of forth-back permutations, as we calledthem. These are the permutations s ∈ Sn with

either s−1(j) < j > s(j) or s−1(j) > j < s(j) (5.2)for all j ∈ 1, 2, . . . , n. Since no forth-back permutation can contain a cycle of oddlength, n must be even for there to exist forth-back permutations, say n = 2m > 0.In that case, we actually have the following lemma:

Lemma 5.1. The number of forth-back permutations in S2m is the Euler zigzagnumber A2m.

Proof. A bijection between the forth-back permutations s and the zigzag permu-tations in S2m is obtained by applying the so-called transformation fundamentale[5]. To perform this transformation, we write s in cycle notation

s = (s1, s2, . . . , sℓ2−1)(sℓ2 , sℓ2+1, . . . , sℓ3−1)(sℓ3 , sℓ3+1, . . . , sℓ4−1) · · ·(sℓm , sℓm+1, . . . , s2m) . (5.3)

This representation and the numbers sj are uniquely determined if we requirethat the first entry of every cycle is bigger than all other entries in that cycle, andalso that s1 < sℓ2 < sℓ3 < · · · < sℓm . The new permutation s is then obtained byforgetting brackets and setting s(j) := sj . We just have to see that this actuallyyields a bijection s 7→ s between forth-back and zigzag permutations. To do thiswe proceed as follows.

Assume first that s is forth-back. Then all cycles necessarily have even lengthand the permutation s is obviously zigzag, s1 > s2 < s3 > s4 < · · · > s2m.


Conversely, let us show that every zigzag permutation s has a unique pre-images, and that that pre-image is forth-back. To construct a pre-image s of s, we onlyneed to find suitable numbers ℓj , which indicate where we have to insert bracketsinto the sequence (s1, s2, . . . , s2m) := (s(1), s(2), . . . , s(2m)) to actually get a pre-image. However, if we have already found ℓ2, ℓ3, . . . , ℓj , then ℓj+1 is necessarilythe first index x with sx > sℓj . Using this, we can construct a pre-image s of s inS2m, and it is uniquely determined. Moreover, if s is zigzag then this constructionensures that s1 and the sℓj are peaks and their neighbors and s2m are valleys.Since also s1 > sℓ2−1, sℓ2 > sℓ3−1, . . . , sℓm > s2m, insertion of brackets before thepeaks ℓj yields forth-back cycles in s.

With the bijection established, it is now clear that there are as many forth-backpermutations as there are zigzag permutations in S2m. This number is the Eulerzigzag number A2m.

The number of forth-back permutations with just one cycle is given by thefollowing lemma, which we present here as we think that it can be helpful infuture research. If Cn denotes the subset of cyclic permutations in Sn, we have thefollowing:

Lemma 5.2. The number of forth-back permutations in C2m is A2m−1.

Proof. The cycle notation s = (s1, s2, . . . , s2m) of cyclic permutations s ∈ S2m

is not uniquely determined, as one may rotate the entries cyclically. It becomesuniquely determined if we require that s2m = 2m. In this case, removal of thelast entry yields a sequence (s1, s2, . . . , s2m−1) that is zagzig (with s1 < s2 ass2m was the biggest entry of s). If we define s ∈ S2m−1 by setting s(j) := sj ,for j = 1, 2, . . . , 2m − 1, we obtain a bijection s 7→ s from the cyclic forth-backpermutations in S2m to the zagzig permutations in S2m−1. Indeed, every zagzigpermutation s in S2m−1 has the cycle s := (s(1), s(2), . . . , s(2m−1), 2m) as uniquepre-image. The existence of this bijection shows that the number of cyclic forth-back permutations in S2m is equal to the number of zagzig permutations in S2m−1,which is A2m−1, as for zigzag permutations.

References1. Bourbaki, N.: Algebra I, Addison-Wesley, Reading MA, 1974.2. Chen, S. and Hudson, R. L.: Some properties of quantum Lévy area in Fock and non-Fock

quantum stochastic calculus, Probability and Mathematical Statistics 33 (2013) 425–434.3. Cockcroft, A. M. and Hudson, R. L.: Quantum mechanical Wiener processes, J. Multivariate

Anal. 7 (1977) 107–128.4. Cohen, P. Beazley, Eyre, T. M. W., and Hudson, R. L.: Higher order Itô product formula

and generators of evolutions and flows, International Journal of Theoretical Physics 34No. 8 (1995) 1481–1486.

5. Foata, D. and Schützenberger, M.: Théorie géom. étrique des polynómes eulériens, LectureNotes in Mathematics 138 (1970).

6. Helmes, K. and Schwane, A.: Lévy’s stochastic area formula in higher dimensions. Journalof functional analysis 54 No. 2 (1983) 177–192.

7. Hoffman, M.: The algebra of multiple harmonic series, Journal of Algebra 194 (1997) 477–495.


8. Hudson, R. L.: Sticky shuffle Hopf algebras and their stochastic representations. in Newtrends in stochastic analysis and related topics, A volume in honour of K. D. Elworthy,(2012) 165–181, World Scientific.

9. Hudson, R. L.: Quantum Lévy area as a quantum martingale limit, in: Quantum probabilityand Related Topics XXIX (2013) 169–188, World Scientific.

10. Hudson, R. L. and Lindsay, J. M.: A non-commutative martingale representation theoremfor non-Fock quantum Brownian motion, Journal of Functional Analysis 61 (1985) 202–221.

11. Hudson, R. L., Schauz, U., and Wu, Y.: Moments of quantum Lévy areas using sticky shuffleHopf algebras, ArXiv 1605.00730.

12. Levin, D. and Wildon, M.: A combinatorial method of calculating the moments of Lévy area,Trans. Amer. Math. Soc. 360 (2008) 6695–6709.

13. Lévy, P.: Le mouvement Brownien plan, Amer. Jour. Math. 62 (1940) 487–550.14. Lévy, P.: Processus Stochastiques et Mouvement Brownien, Gauthier-Villars, Paris, 1948.15. Lévy, P.: Calcul des probabilités-fonctions aléatoires Laplaciennes, C. R. Acad. Sci. Paris

Ser. A-B 229 (1949) 1057–1058.16. Lévy, P.: Calcul des probabilités-sur l’aire comprise entre un arc de la courbe du mouvement

Brownien plan et sa corde, C. R. Acad. Sci. Paris Ser. A-B 230 (1950) 432–434.17. Lévy, P.: Wiener’s random function and other Laplacian functions, in: Proc. 2nd Berkeley

Symposium Math. Statistics and Probability 1950 (1951) 171–187, University of CaliforniaPress.

18. Parthasarathy, K. R.: An Introduction to Quantum Stochastic Calculus, Birkhäuser, 1992.19. Petersen, T. K.: Eulerian Numbers, Birkhäuser, 2015.20. Stanley, R. P.: A survey of alternating permutations, Contemporary Mathematics 531 (2010)

165–196.21. Yor, M.: Remarques sur une Formule de Paul Lévy, in: Séminaire de Probabilités XIV, Lect.

Notes in Math. 784 (1980), Springer-Verlag, Berlin.

Robin Hudson: Loughborough University, Loughborough LE11 3TU, Great BritainE-mail address: [email protected]

Uwe Schauz: Xi’an Jiaotong-Liverpool University, Suzhou, ChinaE-mail address: [email protected]

Yue Wu: Technical University Berlin, Berlin, GermanyE-mail address: [email protected]

ESSENTIAL SETS FOR RANDOM OPERATORS

CONSTRUCTED FROM AN ARRATIA FLOW

A. A. DOROGOVTSEV AND IA. A. KORENOVSKA

Abstract. In this paper we consider a strong random operator Tt whichdescribes a shift of functions from L2(R) along an Arratia flow. We find

a compact set in L2(R) that doesn’t disappear under Tt, and estimate itsKolmogorov widths.

1. Introduction: Arratia Flow and Random Operators

In this paper we consider random operators in L2(R) which describe shifts offunctions along an Arratia flow [1]. Let us recall the definition.

Definition 1.1 ([1]). A family of random processes x(u, s), u ∈ R, s ≥ 0 iscalled an Arratia flow if

1) for each u ∈ R x(u, ·) is a Wiener process with respect to the joint filtrationsuch that x(u, 0) = u;

2) for any u1 ≤ u2 and t ≥ 0

x(u1, t) ≤ x(u2, t) a.s.

3) the joint characteristics are

d < x(u1, ·), x(u2, ·) > (t) = 1Ix(u1,t)=x(u2,t)dt.

In the informal language, Arratia flow is a family of Wiener processes startedfrom each point of R, which move independently up to the meeting, coalesce, andmove together. It was proved in [4, 8] that for any a, b ∈ R and t > 0 the setx([a; b], t) is finite a.s. Since Arratia flow has a right-continuous modification [3],x(·, t) : R → R is a step function for any time t > 0. Hence, for any a, b ∈ R andt > 0 with probability one there exists a random point y ∈ R for which

λu ∈ [a; b] : x(u, t) = y > 0, (1.1)

where λ is Lebesgue measure on R. Since x(·, t) is a right-continuous step function,for a fixed countable set A

Px(R, t) ∩A = ∅ = Px(Q, t) ∩A = ∅

≤∑u∈Q

Px(u, t) ∈ A = 0. (1.2)

Received 2017-7-28; Communicated by the editors.2010 Mathematics Subject Classification. Primary 60H25.

Key words and phrases. Arratia flow, Kolmogorov widths, random operator.

301



302 A. A. DOROGOVTSEV AND IA. A. KORENOVSKA

Since for any a < b the difference x(b,·)−x(a,·)√2

is a Wiener processes until the

collision happens, and x(b,0)−x(a,0)√2

= b−a√2, one can find the distribution of the time

of coalescence τa,b = infs ≥ 0| x(a, s) = x(b, s) of the processes x(a, ·), x(b, ·),i.e. for any t ≥ 0

Pτa,b ≤ t = Px(a, t) = x(b, t)

=

√2

π

∫ ∞

b−a√2t

e−v2

2 dv. (1.3)

Let us notice that for a fixed time t > 0 and an Arratia flow X = x(u, s), u ∈R, s ∈ [0; t] there exists an Arratia flow Y = y(u, r), u ∈ R, r ∈ [0; t] such that

trajectories of X and Y = y(u, t − r), u ∈ R, r ∈ [0; t] don’t cross [1, 7]. Yis called a conjugated (or dual) Arratia flow. It was proved in [10] the followingchange of variable formula for an Arratia flow.

Theorem 1.2 ([10]). For any time t > 0 and nonnegative measurable functionh : R → R such that

∫R h(u)du < ∞∫Rh(x(u, t))du =

∫Rh(u)dy(u, t) a.s., (1.4)

where the last integral is in sense of Lebesgue-Stieltjes.

In this paper we consider random operators Tt, t > 0, in L2(R) which aredefined as follows

(Ttf) (u) = f(x(u, t)),

where f ∈ L2(R) and u ∈ R. It was proved in [5] that Tt is a strong randomoperator [11] in L2(R), but, as it was shown in [10], is not a bounded one. Really,for the point y from (1.1) one can introduce a sequence of the intervals Ai = [ri; pi]such that y ∈ Ai for any i ≥ 1 and pi − ri → 0, i → ∞. Thus for any i ≥ 1

∥Tt1IAi∥2L2(R) ≥ λu ∈ [a; b] : x(u, t) = y > 0,

which can’t be true if Tt was a bounded random operator. Hence, the image ofa compact set under Tt may not be a random compact set. Moreover, as it wasmentioned in [9], the image of a compact set under strong random operator maynot exist. However, in [10] it was presented a family of compact sets in L2(R)whose images under Tt exist and are random compact sets. In this paper weconsider a compact set of this type, and investigate the change of its Kolmogorovwidths [12] under Tt.

2. Tt-essential Functions

If the support of the function f ∈ L2(R) is bounded, suppf ⊂ [a; b], then Ttfequals to 0 with positive probability. Really, by (1.4), one can check that

ESSENTIAL SETS FOR RANDOM OPERATORS 303

P

∞∫

−∞

f2(x(u, t))du = 0

≥ P x(R, t) ∩ [a; b] = ∅

= P

∞∫

−∞

1I[a;b](x(u, t))du = 0

= P

∞∫

−∞

1I[a;b](u)dy(u, t) = 0

,

where y(u, s), u ∈ R, s ∈ [0; t] is a conjugated Arratia flow. Since, by (1.3),

P

∞∫

−∞

1I[a;b](u)dy(u, t) = 0

= P y(b, t) = y(a, t) > 0,

then P∥Ttf∥L2(R) = 0

> 0. This leads to the following definition.

Definition 2.1. For a fixed t > 0 a function f ∈ L2(R) is said to be a Tt-essentialif

P∥Ttf∥L2(R) > 0

= 1.

Example 2.2. Let f ∈ L2(R) be an analytic function which doesn’t equal totallyto zero. Denote the set of its zeroes Zf = u ∈ R| f(u) = 0. Then, by (1.2),P x(R, t) ∩ Zf = ∅ = 1, so f is a Tt-essential for any t > 0.

Let us notice that if t1 = t2 then Tt1-essential function may not be a Tt2-essential. To introduce a T1-essential that is not T2-essential function let us con-sider an increasing sequence uk∞k=0 such that u0 = 0, u1 = 1 and for any n ∈ N

u2n+1 − u2n =1

2n, u2n = u2n−1 + 2n(ln 2)

12 .

Theorem 2.3. The function f =∑∞

n=0 1I[u2n;u2n+1] is a T1-essential, and is nota T2-essential.

Proof. To prove that f is not a T2 essential we show that P ∥T2f∥L2(R) > 0 < 1.Since [u2k;u2k+1] ∩ [u2j ;u2j+1] = ∅ for any k = j then, by (1.4),

P ∥T2f∥2L2(R) > 0 = P

∞∫

−∞

( ∞∑n=0

1I[u2n;u2n+1](x(u, 2))

)2

du > 0

= P

∞∑

n=0

∞∫−∞

1I[u2n;u2n+1](x(u, 2))du > 0


= P

∞∑n=0

(y(u2n+1, 2)− y(u2n, 2)) > 0

= P ∃n ≥ 0 : y(u2n+1, 2) = y(u2n, 2)

≤∞∑

n=0

P y(u2n+1, 2) = y(u2n, 2) .

Thus by (1.3),

∞∑n=0

P y(u2n+1, 2) = y(u2n, 2) =∞∑

n=0

1√4π

1

2n+1∫− 1

2n+1

e−v2

4 dv ≤ 1√π

< 1.

Consequently, the function f =∑∞

n=0 1I[u2n;u2n+1] is not a T2-essential. To prove

that f =∑∞

n=0 1I[u2n;u2n+1] is a T1-essential one can show the following estimation.

Lemma 2.4. Let w(un, ·)∞n=0 be a family of independent Wiener processes on[0; 1] such that w(un, 0) = un. Then for any n ∈ N

P

maxs∈[0;1]

maxj=0,2n−1

w(uj , s) ≥ mins∈[0;1]

w(u2n, s)

<

1

2n2√π ln 2

.

Proof. Let w1, w2 be an independent Wiener processes on [0; 1] started from point0, i.e. w1(0) = w2(0) = 0. It can be noticed that

P

maxs∈[0;1]

maxj=0,2n−1

w(uj , s) ≥ mins∈[0;1]

w(u2n, s)

= P

∃ j = 0, 2n− 1 : max

s∈[0;1]w(uj , s)− min

s∈[0;1]w(u2n, s) ≥ 0

≤

2n−1∑j=0

P

maxs∈[0;1]

w(uj , s)− mins∈[0;1]

w(u2n, s) ≥ 0

≤2n−1∑j=0

P

maxs∈[0;1]

w1(s)− mins∈[0;1]

w2(s) ≥ u2n − uj

.

From the fact that un∞n=0 is an increasing sequence we can estimate the lastexpression and complete the proof

2n−1∑j=0

P

maxs∈[0;1]

w1(s)− mins∈[0;1]

w2(s) ≥ u2n − uj

≤ 1√π

2n−1∑j=0

1

u2n − uje−

(u2n−uj)2

4

≤ 2n− 1√π(u2n − u2n−1)

e−(u2n−u2n−1)2

4

≤ 1

2n2√π ln 2

.


Let us prove that the function f =∑∞

n=0 1I[u2n;u2n+1] is a T1-essential. Using thereasoning from the first part of the proof it can be checked that for the consideredfunction f the following equality holds

P ∥T1f∥L2(R) > 0 = P

∞∑n=0

(y(u2n+1, 1)− y(u2n, 1)) > 0

.

Let us prove that

P

lim supn→∞

(y(u2n+1, 1)− y(u2n, 1)) ≥ 1

= 1. (2.1)

Build a new processes y(un, ·)∞n=0 such that y(un, ·)∞n=0 and y(un, ·)∞n=0 havethe same distributions in C ([0; 1])

∞in the following way [4]. Let w(un, ·)∞n=0 be

a given family of Wiener processes on [0; 1], w(un, 0) = un. Let us denote collisiontime of f, g ∈ C ([0; 1]) by τ [f, g] := inft | f(t) = g(t). Put y(u0, ·) := w(u0, ·).Then for any n ∈ N, s ∈ [0; 1] one can define

y(un, s) = w(un, s)1I s < τ [w(un, ·), y(un−1, ·)] + y(un−1, s)1Is ≥ τ [w(un, ·), y(un−1, ·)] .

According to constructions of stochastic processes y(un, ·)∞n=0

P ∃ N ∈ N : ∀ n ≥ N y(u2n, t) = w(u2n, t),

y(u2n+1, t) = w(u2n+1, t)1I t < τ [w(u2n, ·), w(u2n+1, ·)]+w(u2n, t)1I t ≥ τ [w(u2n, ·), w(u2n+1, ·)] = 1. (2.2)

Thus

P ∃ N ∈ N : ∀ n ≥ N y(u2n+1, t)− y(u2n, t) = w(u2n+1, t)− w(u2n, t) = 1.

For the considered sequence un∞n=0 and any n ∈ N the following inequality holds

P w(u2n+1, t)− w(u2n, t) ≥ 1 =

∞∫1

1√4π

e−(v− 1

2k)2

4 dv ≥ 1√4π

∞∫1

e−v2

4 dv.

Therefore, by the Borel-Cantelli lemma and (2.2),

P lim supn→∞

(y(u2n+1, t)− y(u2n, t)) ≥ 1 = 1.

Using the observation from Example 2.2 one can introduce a family of Tt-essential functions for all t > 0.

For any ε > 0 let us consider an integral operator Kε in L2(R) with the kernel

kε(v1, v2) =

∫Rpε(u− v1)pε(u− v2)dy(u, t), (2.3)


where v1, v2 ∈ R, and pε(u) =1√2πε

e−u2

2ε . By the change of variables formula for

an Arratia flow [10],

(Kεf, f) =

∫R(f ∗ pε)2(x(u, t))du. (2.4)

Lemma 2.5. For any ε > 0 and nonzero function f ∈ L2(R)

P (Kεf, f) = 0 = 1.

Proof. According to (2.1) it is sufficient to note that f ∗pε is an analytic function.Consequently, for any t > 0 the following relations are true

P (K1f, f) > 0 = P∥Tt(f ∗ p1)∥L2(R) > 0

= P x(R, t) ∩ Zf∗p1 = ∅ = 1.

According to the last theorem and (2.4), for any ε > 0 and nonzero f ∈ L2(R)the function f ∗ pε is a Tt-essential for each t > 0.

3. Change of Compact Sets under a Strong Random OperatorGenerated by an Arratia Flow

As it was noticed in the introduction any function with bounded support isn’ta Tt-essential. Consequently, if K ⊆ L2(R) is a compact set of functions withuniformly bounded supports such that Tt(K) is well-defined, then the image Tt(K)equals to 0 with positive probability. It was shown in [10] that Tt may alsochange the geometry of K even in the case of a compact set K for which Tt(K) =0 a.s. For example, the image Tt(K) of a convergent sequence and its limitingpoint may not have limiting points. In this section we build a compact set K forwhich Tt(K) = 0 a.s. and investigate the change of its Kolmogorov-widths inL2(R) under random operator Tt.

Definition 3.1 ([12]). The Kolmogorov n-width of a set C ⊆ H in a Hilbert spaceH is given by

dn(C) = infdimL≤n

supf∈C

infg∈L

∥f − g∥H ,

where L is a subspace of H.

We consider the following compact set in L2(R)

K =f ∈ W 1

2 (R)|∫Rf2(u)(1 + |u|)3du+

∫R(f ′(u))

2(1 + |u|)7du ≤ 1

. (3.1)

Estimations on its Kolmogorov-widths in L2(R) are presented in the next lemma.

Lemma 3.2. There exist positive constants C1, C2 such that for any n ∈ NC1

n≤ dn (K) ≤ C2

n310

.


Proof. Let n ∈ N be fixed. To estimate dn (K) from above one can consider the

partition uknk=0 of [−n15 ;n

15 ] into n segments [uk;uk+1], k = 0, n− 1 with

equal lengths. Let us show that for the n-dimensional subspace

Ln = LS1I[uk;uk+1], k = 0, n− 1

supf∈K

infg∈Ln

∥f − g∥L2(R) ≤C2

n310

.

If f ∈ K then∫R f2(u)(1 + |u|)3du ≤ 1. Thus for any C > 0∫

|u|>c

f2(u)du ≤ 1

(1 + C)3

∫|u|>c

f2(u)(1 + |u|)3du ≤ 1

C3.

So, for the function gf =n−1∑k=0

f(uk)1I[uk;uk+1] ∈ Ln the following estimation is true

∥f − gf∥2L2(R) ≤1

n35

+

∫|u|≤n

15

(f(u)− gf (u))2du.

By the Cauchy inequality, for f ∈ K and u ∈ [uk;uk+1] u∫uk

f ′(v)dv

2

≤u∫

uk

dv

(1 + |v|)7≤ u− uk.

Consequently, ∫|u|≤n

15

(f(u)− gf (u))2du =

n−1∑k=0

uk+1∫uk

u∫uk

f ′(v)dv

2

du

≤ 1

2

n−1∑k=0

(uk+1 − uk)2 =

2

n35

,

and the upper estimation for dn(K) holds with the constant C2 = 312 .

To get a lower estimation for dn(K) we use the theorem about n-width of (n+1)-

dimensional ball [12]. Let uk2(n+1)k=0 be a partition of [0; 1] into 2(n+1) segments

[uk;uk+1], k = 0, 2n+ 1 with equal lengths. Consider (n+1)-dimensional spaceLn+1 = LSfk, k = 0, n, where the functions fk, k = 0, n, are defined as follows

fk =

0, u /∈ [u2k;u2k+1],

1, u ∈ [u2k + 16(n+1) ;u2k + 2

6(n+1) ],

6(n+ 1)(u− u2k), u ∈ [u2k;u2k + 16(n+1) ],

−6(n+ 1)(u− u2k+1), u ∈ [u2k + 26(n+1) ;u2k+1].

(3.2)

We show that if c = 23(5+29·33)5 then the ball Bn+1 = f ∈ Ln+1| ∥f∥L2(R) ≤

1√cn is a subset of K. Since ∥fk∥2L2(R) =

518(n+1) , k = 0, n, then for any f ∈ Bn+1

such that f =n∑

k=0

ckfk the following relation holds∑n

k=0 c2k ≤ 36

5cn . Thus according


to (3.2),∫R

f2(u)(1 + |u|)3du+

∫R

(f ′(u))2(1 + |u|)7du

≤ 23∥f∥2L2(R) + 27 ·n∑

k=0

c2k

u2k+

16(n+1)∫

u2k

(6(n+ 1))2du+

u2k+1∫u2k+

26(n+1)

(6(n+ 1))2du

≤ 23

cn2+ 210 · 3n 36

5cn≤ 1

c· 2

3(5 + 29 · 33)5

= 1.

Consequently, Bn+1 ⊂ K and dn(K) ≥ dn(Bn+1). Due to the theorem aboutn-width of (n+1)-dimensional ball, dn(Bn+1) =

1√cn

[12]. So the lower estimation

for dn(K) holds with C1 :=√c.

To show that estimations from above for the Kolmogorov-widths of the con-sidered compact set K don’t change under Tt one may use the same idea as inLemma 2.

Theorem 3.3. There exists Ω of probability one such that for any ω ∈ Ω andn ∈ N

dn (Tωt (K)) ≤ C(ω)

n310

, (3.3)

where the constant C(ω) > 0 doesn’t depend on n.

Proof. For a fixed n ∈ N let us consider a partition uknk=0 of [−n15 ;n

15 ] into

n segments with equal lengths. To prove (3.3) it’s sufficient to show the follow-ing inequality for the linear space Lω

n = LS Tωt 1I[uk;uk+1], k = 0, n− 1 with

dimension at most n

suph1∈Tω

t (K)

infh2∈Lω

n

∥h1 − h2∥L2(R) ≤C(ω)

n310

.

According to the change of variable formula for an Arratia flow, one can check theequality for any f ∈ K∥∥∥∥∥Tω

t f − Tωt

(n−1∑k=0

f(uk)1I[uk;uk+1]

)∥∥∥∥∥2

L2(R)

=

∫|u|>n

15

f2(u)dy(u, t, ω)

+

∫|u|≤n

15

(f(u)−

n−1∑k=0

f(uk)1I[uk;uk+1](u)

)2

dy(u, t, ω).


To estimate from above the last integral let us notice that

∫|u|≤n

15

(f(u)−

n−1∑k=0

f(uk)1I[uk;uk+1](u)

)2

dy(u, t, ω)

≤n−1∑k=0

∫ uk+1

uk

(∫ u

uk

|f ′(v)| dv)2

dy(u, t, ω).

Due to (3.1), for any f ∈ K and u ∈ [uk;uk+1](∫ u

uk

|f ′(v)| dv)2

≤∫ u

uk

dv

(1 + |v|)7≤ uk+1 − uk.

Thus

n−1∑k=0

∫ uk+1

uk

(∫ u

uk

|f ′(v)| dv)2

dy(u, t, ω) ≤n−1∑k=0

(uk+1 − uk)

∫ uk+1

uk

dy(u, t, ω)

=2

n45

(y(n15 , t, ω)− y(−n

15 , t, ω)).

For an Arratia flow y(u, s), u ∈ R, s ∈ [0; t] the following relation is true [2]

lim|u|→∞

|y(u, t)||u|

= 1 a.s.

Consequently, for any ω ∈ Ω = ω′ ∈ Ω| lim|u|→∞

|y(u,t,ω′)||u| = 1 the estimation holds

∫|u|≤n

15

(f(u)−

n−1∑k=0

f(uk)1I[uk;uk+1](u)

)2

dy(u, t, ω) ≤ 4c(ω)

n35

(3.4)

with the constant

c(ω) = sup|u|≥1

|y(u, t, ω)||u|

. (3.5)

Let us prove that for any ω ∈ Ω there exists a constant c(ω) such that∫|u|>n

15

f2(u)dy(u, t, ω) ≤ c(ω)

n35

.

It can be noticed that∫|u|>n

15f2(u)dy(u, t) ≤ 1

n35

∫|u|>n

15f2(u)(1 + |u|)3dy(u, t).

Denote by θj∞j=1 a sequence of jump points of the function y(·, t) on R+. Thus


one may show∫u>n

15

f2(u)(1 + u)3dy(u, t) =∑

θi≥n15

f2(θi)(1 + θi)3∆y(θi, t)

=

∞∑k=1

∑i: θi∈[k;k+1)

f2(θi)(1 + θi)3∆y(θi, t)

≤∞∑k=1

(2 + k)3∑

i: θi∈[k;k+1)

f2(θi)∆y(θi, t).

According to the Cauchy inequality and (3.1), for any u ∈ R+ the following rela-tions hold

f2(u) ≤∫ ∞

u

(f ′(v))2(1 + v)7dv ·

∫ ∞

u

dv

(1 + v)7≤ 1

6u6.

Consequently, due to (3.5), the inequalities are true

∞∑k=1

(2 + k)3∑

i: θi∈[k;k+1)

f2(θi)∆y(θi, t)

≤∞∑k=1

(2 + k)31

6k6(y(k + 1, t)− y(k, t))

≤ 16c

3

∞∑k=1

1

k2.

Hence, for any ω ∈ Ω there exists the constant C1(ω) =16c(ω)

3 such that∫u>n

15

f2(u)dy(u, t, ω) ≤ C1(ω)

n35

.

Similarly, it can be proved that∫

u<−n15

f2(u)dy(u, t, ω) ≤ C1(ω)

n35

. According to this

and (3.4), for any ω ∈ Ω an upper estimation for dn (Tωt (K)) is true.

The functions from Lemma 2 that were used to build the (n + 1)-dimensionalsubspace are not Tt-essential for any t > 0. Thus the image of this subspace underthe random operator Tt may be equal to 0 with positive probability. So, one canask about the existence of a finite-dimensional subspace such that for any t > 0its image under Tt is a linear subspace with the same dimension.

4. A Subspace Preserving the Dimension under a RandomOperator Generated by an Arratia Flow

In this section for any t > 0 and n ∈ N we present a family gk, k = 0, nof linearly independent Tt-essential functions such that their images under Tt arelinearly independent. Such a family generates a subspace which preserves the


dimension under a random operator generated by an Arratia flow. It can be usedto get a lower estimation of dn(Tt(K)).

Let us fix any n ∈ N, and build a family of (n+1) linearly independent functions

in the following way. Let uk2(n+1)k=0 be a partition of [0;n−2] into 2(n+1) segments

with equal lengths. For any k = 0, n define fk by

fk =

0, u /∈ [u2k;u2k+1],

1, u ∈ [u2k + n−2

6(n+1) ;u2k + 2n−2

6(n+1) ],6(n+1)n−2 (u− u2k), u ∈ [u2k;u2k + n−2

6(n+1) ],

− 6(n+1)n−2 (u− u2k+1), u ∈ [u2k + 2n−2

6(n+1) ;u2k+1].

(4.1)

Lemma 4.1. There exists ε0 > 0 such that for any 0 < ε ≤ ε0 the functionsfk ∗ pε, k = 0, n are linearly independent.

Proof. Since the considered functions fk, k = 0, n are linearly independent, itsGram determinant doesn’t equal to 0, i.e. G(f0, . . . , fn) = 0. For each k = 0, n

fk ∗ pε → fk, ε → 0.

Hence, due to the continuity of the Gram determinant, one may notice that thereexists ε0 > 0 such that for any 0 < ε ≤ ε0

G(f0 ∗ pε, . . . , fn ∗ pε) = 0,

and the desired result is proved.

Theorem 4.2. There exists a set Ω0 of probability one such that for any ω ∈ Ω0

the functions Tωt (f0 ∗ pε), . . . , Tω

t (fn ∗ pε) are linearly independent.

Proof. Denote by Kε the integral operator in L2(R) with the kernel kε. Toprove the statement of the theorem it’s enough to show that on some Ω0 ofprobability one the following inequality holds (Kεf, f) > 0, for any nonzerof ∈ LSf0, . . . , fn. Due to (1.4)

(Kεf, f) =∑θ

(f ∗ pε)2(θ)∆y(θ, t), (4.2)

where θ is a point of jump of the function y(·, t).It was proved in [6] that there exists Ω0 of probability one such that for any

ω ∈ Ω0 a linear span of the functions pε(· − θ(ω))|[0;1]θ(ω) is dense in L2([0; 1]).Thus on the set Ω0 for any f ∈ LSf0, . . . , fn ⊂ L2([0; 1]) one can find a randompoint θf such that (f(·), pε(· − θf )) = 0. Since y(·, t) : R → R is nondecreasing,∆y(θ, t) > 0 for any jump-point θ. Consequently, on the set Ω0∑

θ

(f ∗ pε)2(θ)∆y(θ, t) =∑θ

(f(·), pε(· − θ))2∆y(θ, t)

≥ (f(·), pε(· − θf ))2∆y(θf , t) > 0,

which proves the theorem.


Acknowledgment. The first author acknowledges financial support from theDeutsche Forschungsgemeinschaft (DFG) within the project ”Stochastic Calcu-lus and Geometry of Stochastic Flows with Singular Interaction” for initiation ofinternational collaboration between the Institute of Mathematics of the Friedrich-Schiller University Jena (Germany) and the Institute of Mathematics of the Na-tional Academy of Sciences of Ukraine, Kiev.

The second author is grateful to the Institute of Mathematics of the Friedrich-Schiller University Jena (Germany) for hospitality and financial support withinErasmus+ programme 2015/16-2017/18 between Jena Friedrich-Schiller University(Germany) and the Institute of Mathematics of the National Academy of Sciencesof Ukraine, Kiev.

References

1. Arratia, R.: Coalescing Brownian motions on the line, PhD Thesis, University of Wisconsin,Madison, 1979.

2. Chernega, P. P.: Local time at zero for Arratia flow, Ukr. Mat. Zh. 64 (2014), no. 4, 542–556.

3. Dorogovtsev, A. A.: Some remarks about Brownian flow with coalescence, Ukr. Mat. Zh. 57(2005), no. 10, 1327–1333.

4. Dorogovtsev, A. A.: Measure-valued processes and stochastic flows (in Russian), Instituteof Mathematics of the NAS of Ukraine, Kiev, 2007.

5. Dorogovtsev, A. A.: Krylov-Veretennikov expansion for coalescing stochastic flows, Com-mun. Stoch. Anal. 6 (2012), no. 3, 421–435.

6. Dorogovtsev, A. A. and Korenovska, Ia. A.: Some random integral operators related to apoint process, in: Theory of Stochastic Processes. 22(38) (2017), no. 1.

7. Fontes, L. R. G., Isopi, M., Newman, C. M., and Ravishankar, K.: The Brownian web, Proc.Nat. Acad. Sciences. 99 (2002), no. 25, 15888–15893.

8. Harris, T. E.: Coalescing and noncoalescing stochastic flows in R1, Stochastic Processes and

their Applications 17 (1984), no. 2, 187–210.9. Korenovska, I. A.: Random maps and Kolmogorov widths, Theory of Stochastic Processes

20(36) (2015), no. 1, 78–83.10. Korenovska, Ia. A.: Properties of strong random operators generated by an Arratia flow (in

Russian), Ukr. Mat. Zh. 69 (2017), no. 2, 157–172.11. Skorokhod, A. V.: Random Linear Operators, D.Reidel Publishing Company, Dordrecht,

Holland, 1983.12. Tikhomirov, V. M.: Some questions in approximation theory (in Russian), Izdat. Moskov.

Univ., Moscow, 1976.

A. A. Dorogovtsev: Institute of Mathematics, National Academy of Sciences ofUkraine, Kiev, Ukraine, and National Technical University of Ukraine “Igor SikorskyKyiv Polytechnic Institute, Institute of Physics and Technology, Kiev, Ukraine.

E-mail address: [email protected]

Ia. A. Korenovska: Institute of Mathematics, National Academy of Sciences ofUkraine, Kiev, Ukraine


A NOTE ON TIME-DEPENDENT ADDITIVE FUNCTIONALS

ADRIEN BARRASSO AND FRANCESCO RUSSO

Abstract. This note develops shortly the theory of non-homogeneous ad-

ditive functionals and is a useful support for the analysis of time-dependentMarkov processes and related topics. It is a significant tool for the analysisof Markovian BSDEs in law. In particular we extend to a non-homogeneoussetup some results concerning the quadratic variation and the angular bracket

of Martingale Additive Functionals (in short MAF) associated to a homoge-neous Markov processes.

1. Introduction

The notion of Additive Functional of a general Markov process is due to E.BDynkin and has been studied since the early ’60s by the Russian, French andAmerican schools of probability, see for example [4, 8, 16]. A mature version ofthe homogeneous theory may be found for example in [7], Chapter XV. In thatcontext, given an element x in some state space E, Px denotes the law of a time-homogeneous Markov process with initial value x.

An additive functional is a right-continuous process (At)t≥0 defined on a canon-ical space, adapted to the canonical filtration such that for any s ≤ t and x ∈ E,As+t = As + At θs Px-a.s., where θ is the usual shift operator on the canonicalspace. If moreover A is under any law Px a martingale, then it is called a Martin-gale Additive Functional (MAF). The quadratic variation and angular bracket of aMAF were shown to be AFs in [7]. We extend this type of results to a more generaldefinition of an AF which is closer to the original notion of Additive Functionalassociated to a stochastic system introduced by E.B. Dynkin, see [9] for instance.

Our setup is as follows. We consider a canonical Markov class (Ps,x)(s,x)∈[0,T ]×Ewith time index [0, T ] and state space E being a Polish space. For any (s, x) ∈[0, T ] × E, Ps,x corresponds to the probability law (defined on some canonicalfiltered space

(Ω,F , (Ft)t∈[0,T ]

)) of a Markov process starting from point x at

time s. On (Ω,F), we define a non-homogeneous Additive Functional (shortenedby AF) as a real-valued random-field A := (Atu)0≤t≤u≤T verifying the two followingconditions.

(1) For any 0 ≤ t ≤ u ≤ T , Atu is Ft,u-measurable;(2) for any (s, x) ∈ [0, T ]× E, there exists a real cadlag (Fs,x

t )t∈[0,T ]-adaptedprocess As,x (taken equal to zero on [0, s] by convention) such that for anyx ∈ E and s ≤ t ≤ u, Atu = As,xu −As,xt Ps,x a.s.

Received 2017-8-8; Communicated by the editors.2010 Mathematics Subject Classification. Primary 60J55; 60J35; 60G07; 60G44.

Key words and phrases. Additive functionals, Markov processes, covariation.

313



314 ADRIEN BARRASSO AND FRANCESCO RUSSO

Ft,u denotes the σ-field generated by the canonical process between time t and u,and Fs,x

t is obtained by adding the Ps,x negligible sets to Ft. As,x will be calledthe cadlag version of A under Ps,x. If for any (s, x), As,x is a (Ps,x, (Ft)t∈[0,T ])-square integrable martingale then A will be called a square integrable MartingaleAdditive Functional (in short, square integrable MAF).

The main contributions of the paper are essentially the following. In Section 3,we recall the definition and prove some basic results concerning canonical Markovclasses. In Section 4, we start by defining an AF in Definition 4.1. In Proposi-tion 4.4, we show that if (M t

u)0≤t≤u≤T is a square integrable MAF, then thereexists an AF ([M ]tu)0≤t≤u≤T which for any (s, x) ∈ [0, T ]×E, has [Ms,x] as cadlagversion under Ps,x. Corollary 4.11 states that given two square integrable MAFs(M t

u)0≤t≤u≤T , (Ntu)0≤t≤u≤T , there exists an AF, denoted by (⟨M,N⟩tu)0≤t≤u≤T ,

which has ⟨Ms,x, Ns,x⟩ as cadlag version under Ps,x. Finally, we prove in Propo-sition 4.17 that if M or N is such that for Ps,x, its cadlag version under Ps,x,its angular bracket is absolutely continuous with respect to some continuous non-decreasing function V , then there exists a Borel function v such that for any (s, x),

⟨Ms,x, Ns,x⟩ =∫ ·∨ss

v(r,Xr)dVr.The present note constitutes a support for the authors, in the analysis of de-

terministic problems related to Markovian type backward stochastic differentialequations where the forward process is given in law, see e.g. [2]. Indeed, when theforward process of the BSDE does not define a stochastic flow (typically if it is notthe strong solution of an SDE but only a weak solution), we cannot exploit thementioned flow property to show that the solution of the BSDE is a function ofthe forward process, as it is usually done, see Remark 5.35 (ii) in [17] for instance.

2. Preliminaries

The present section is devoted to fix some basic notions, notations and vocab-ulary. A topological space E will always be considered as a measurable spacewith its Borel σ-field which shall be denoted B(E) and if S is another topologi-cal space equipped with its Borel σ-field, B(E,S) (resp. Bb(E,S), resp. C(E,S),resp. Cb(E,S)) will denote the set of Borel (resp. bounded Borel, reps. continu-ous, resp. bounded continuous) functions from E to S. Let T ∈ R∗

+, d ∈ N∗, then

C1,2b ([0, T ]×Rd) will denote the space of bounded continuous real valued functions

on [0, T ] × Rd which are differentiable in the first variable, twice differentiable inthe second with bounded continuous partial derivatives.

Let (Ω,F), (E, E) be two measurable spaces. A measurable mapping from(Ω,F) to (E, E) shall often be called a random variable (with values in E), or inshort r.v. If T is some set, an indexed set of r.v. with values in E, (Xt)t∈T willbe called a random field (indexed by T with values in E). In particular, if T is aninterval included in R+, (Xt)t∈T will be called a stochastic process (indexed by Twith values in E). Given a stochastic process, if the mapping

(t, ω) 7−→ Xt(ω)(T× Ω,B(T)⊗F) −→ (E, E)

is measurable, then the process (Xt)t∈T will be called a measurable process (in-dexed by T with values in E).

A NOTE ON TIME-DEPENDENT ADDITIVE FUNCTIONALS 315

Let (Ω,F ,P) be a fixed probability space. For any p ≥ 1, Lp := Lp(R) willdenote the set of real valued random variables with finite p-th moment. Tworandom fields (or stochastic processes) (Xt)t∈T, (Yt)t∈T indexed by the same setand with values in the same space will be said to be modifications (or versions)of each other if for every t ∈ T, P(Xt = Yt) = 1. If the probability space isequipped with a right-continuous filtration, then (Ω,F , (Ft)t∈T,P) will be calledstochastic basis and will be said to fulfill the usual conditions if the probabilityspace is complete and if F0 contains all the P-negligible sets.

Concerning spaces of real valued stochastic processes on the above mentionedstochastic basis, M will be the space of cadlag martingales. For any p ∈ [1,∞]Hp will denote the subset of M of elements M such that sup

t∈T|Mt| ∈ Lp and in

this set we identify indistinguishable elements. Hp is a Banach space for the norm

∥M∥Hp = E[|supt∈T

Mt|p]1p , and Hp

0 will denote the Banach subspace of Hp whose

elements start at zero.A crucial role in the present note, as well as in classical stochastic analysis

is played by localization via stopping times. If T = [0, T ] for some T ∈ R∗+, a

stopping time will be intended as a random variable with values in [0, T ] ∪ +∞such that for any t ∈ [0, T ], τ ≤ t ∈ Ft. We define a localizing sequenceof stopping times as an increasing sequence of stopping times (τn)n≥0 such thatthere exists N ∈ N for which τN = +∞. Let Y be a process and τ a stoppingtime, we denote Y τ the process t 7→ Yt∧τ which we call stopped process. If C isa set of processes, we define its localized class Cloc as the set of processes Y suchthat there exists a localizing sequence (τn)n≥0 such that for every n, the stoppedprocess Y τn belongs to C.

We say some words about the concept of bracket related to two processes: thesquare bracket and the angular bracket. They coincide if at least one of the twoprocesses is continuous. For any M,N ∈ M [M,N ] denotes the covariation ofM,N . If M = N , we write [M ] := [M,N ]. [M ] is called quadratic variationof M . If M,N ∈ H2

loc, ⟨M,N⟩ (or simply ⟨M⟩ if M = N) will denote their(predictable) angular bracket. H2

0 will be equipped with scalar product definedby (M,N)H2 := E[MTNT ] = E[⟨M,N⟩T ] which makes it a Hilbert space. Twoelements M,N of H2

0,loc will be said to be strongly orthogonal if ⟨M,N⟩ = 0.

If A is an adapted process with bounded variation then V ar(A) (resp. Pos(A),Neg(A)) will denote its total variation (resp. positive variation, negative varia-tion), see Proposition 3.1, chap. 1 in [15]. In particular for almost all ω ∈ Ω,t 7→ V art(A(ω)) is the total variation function of the function t 7→ At(ω).

For more details concerning these notions, one may consult [18] or [15] forexample.

3. Markov Classes

We recall here some basic definitions and results concerning Markov processes.For a complete study of homogeneous Markov processes, one may consult [7],concerning non-homogeneous Markov classes, our reference was Chapter VI of[10].


3.1. Definition and basic results. The first definition refers to the canonicalspace that one can find in [14], see paragraph 12.63.

Notation 3.1. In the whole section E will be a fixed Polish space (a separablecompletely metrizable topological space), and B(E) its Borel σ-field. E will becalled the state space.

We consider T ∈ R∗+. We denote Ω := D(E) the Skorokhod space of functions

from [0, T ] to E right-continuous with left limits and continuous at time T (e.g.cadlag). For any t ∈ [0, T ] we denote the coordinate mapping Xt : ω 7→ ω(t), andwe introduce on Ω the σ-field F := σ(Xr|r ∈ [0, T ]).

On the measurable space (Ω,F), we introduce the canonical process

X :(t, ω) 7−→ ω(t)

([0, T ]× Ω,B([0, T ])⊗F) −→ (E,B(E)),(3.1)

and the right-continuous filtration (Ft)t∈[0,T ] where Ft :=∩

s∈]t,T ]

σ(Xr|r ≤ s) if

t < T , and FT := σ(Xr|r ∈ [0, T ]) = F .(Ω,F , (Ft)t∈[0,T ]

)will be called the canonical space (associated to T and E).

For any t ∈ [0, T ] we denote Ft,T := σ(Xr|r ≥ t), and for any 0 ≤ t ≤ u < T wewill denote Ft,u :=

∩n≥0

σ(Xr|r ∈ [t, u+ 1n ]).

Remark 3.2. All the results of the present paper remain valid if Ω is the space ofcontinuous functions from [0, T ] to E, and if the time index is equal to R+.

We recall that since E is Polish, then D(E) can be equipped with a Skorokhoddistance which makes it a Polish metric space (see Theorem 5.6 in Chapter 3 of[11]), and for which the Borel σ-field is F (see Proposition 7.1 in Chapter 3 of [11]).This in particular implies that F is separable, as the Borel σ-field of a separablemetric space.

Remark 3.3. The above σ-fields fulfill the properties below.

(1) For any 0 ≤ t ≤ u < T , Ft,u = Fu ∩ Ft,T ;(2) for any t ≥ 0, Ft ∨ Ft,T = F ;(3) for any (s, x) ∈ [0, T ]×E, the two first items remain true when considering

the Ps,x-closures of all the σ-fields;(4) for any t ≥ 0, Π := F = Ft ∩ F tT |(Ft, F tT ) ∈ Ft × Ft,T is a π-system

generating F , i.e. it is stable with respect to the intersection.

Definition 3.4. The function

P :(s, t, x, A) 7−→ Ps,t(x,A)

[0, T ]2 × E × B(E) −→ [0, 1],

will be called transition kernel if, for any s, t in [0, T ], x ∈ E, A ∈ B(E), it verifiesthe following.

(1) Ps,t(·, A) is Borel,(2) Ps,t(x, ·) is a probability measure on (E,B(E)),(3) if t ≤ s then Ps,t(x,A) = 1A(x),(4) if s < t, for any u > t,

∫EPs,t(x, dy)Pt,u(y,A) = Ps,u(x,A).


The latter statement is the well-known Chapman-Kolmogorov equation.

Definition 3.5. A transition kernel P is said to be measurable in time if for everyt ∈ [0, T ] and A ∈ B(E), (s, x) 7−→ Ps,t(x,A) is Borel.

Remark 3.6. Let P be a transition kernel which is measurable in time, let ϕ ∈B(E,R) and t ∈ [0, T ]. Assume that for any (s, x) ∈ [0, T ] × E, the integral∫|ϕ|(y)Ps,t(x, dy) exists. Then the mapping (s, x) 7→

∫ϕ(y)Ps,t(x, dy) is Borel.

This can be easily shown by approximating ϕ by simple functions and using thedefinition.

Definition 3.7. A canonical Markov class associated to a transition kernel P isa set of probability measures (Ps,x)(s,x)∈[0,T ]×E defined on the measurable space(Ω,F) and verifying for any t ∈ [0, T ] and A ∈ B(E)

Ps,x(Xt ∈ A) = Ps,t(x,A), (3.2)

and for any s ≤ t ≤ u

Ps,x(Xu ∈ A|Ft) = Pt,u(Xt, A) Ps,x a.s. (3.3)

The statement below comes Formula 1.7 in Chapter 6 of [10].

Proposition 3.8. For any (s, x) ∈ [0, T ]× E, t ≥ s and F ∈ Ft,T yields

Ps,x(F |Ft) = Pt,Xt(F ) = Ps,x(F |Xt) Ps,xa.s. (3.4)

Property (3.4) is often called Markov property. We recall here the concept ofhomogeneous canonical Markov classes and its links with Markov classes.

Notation 3.9. A mapping

P :E × [0, T ]× B(E) −→ [0, 1]

(t, x, A) 7−→ Pt(x,A),(3.5)

will be called a homogeneous transition kernel if

P : (s, t, x,A) 7−→ Pt−s(x,A)1s<t + 1A(x)1s≥t

is a transition kernel in the sense of Definition 3.4. This in particular impliesP = P0,·(·, ·).

A set of probability measures (Px)x∈E on the canonical space associated to Tand E (see Notation 3.1) will be called a homogeneous canonical Markov class

associated to a homogeneous transition kernel P if∀t ∈ [0, T ] ∀A ∈ B(E) ,Px(Xt ∈ A) = Pt(x,A)

∀0 ≤ t ≤ u ≤ T ,Px(Xu ∈ A|Ft) = Pu−t(Xt, A) Ps,xa.s.(3.6)

Given a homogeneous canonical Markov class (Px)x∈E associated to a homoge-

neous transition kernel P , one can always consider the canonical Markov class(Ps,x)(s,x)∈[0,T ]×E associated to the transition kernel

P : (s, x, t, A) 7−→ Pt−s(x,A)1s<t + 1A(x)1s≥t.

In particular, for any x ∈ E, we have P0,x = Px.


For the rest of this section, we are given a canonical Markov class

(Ps,x)(s,x)∈[0,T ]×E

whose transition kernel is measurable in time. Proposition A.10 in [3] states thefollowing.

Proposition 3.10. For any event F ∈ F , (s, x) 7−→ Ps,x(F ) is Borel. For anyrandom variable Z, if the function (s, x) 7−→ Es,x[Z] is well-defined (with possiblevalues in [−∞,∞]), then it is Borel.

Definition 3.11. For any (s, x) ∈ [0, T ]×E we will consider the (s, x)-completion(Ω,Fs,x, (Fs,x

t )t∈[0,T ],Ps,x)

of the stochastic basis(Ω,F , (Ft)t∈[0,T ],P

s,x)

bydefining Fs,x as the Ps,x-completion of F , by extending Ps,x to Fs,x and finallyby defining Fs,x

t as the Ps,x-closure of Ft, for every t ∈ [0, T ].

We remark that, for any (s, x) ∈ [0, T ] × E,(Ω,Fs,x, (Fs,x

t )t∈[0,T ],Ps,x)is a

stochastic basis fulfilling the usual conditions, see 1.4 in [15] Chapter I.We recall the following simple consequence of Remark 32 in [5] Chapter II.

Proposition 3.12. Let G be a sub-σ-field of F , P a probability on (Ω,F) and GPthe P-closure of G. Let ZP be a real GP-measurable random variable. There existsa G-measurable random variable Z such that Z = ZP P-a.s.

From this we can deduce the following.

Proposition 3.13. Let (s, x) ∈ [0, T ] × E be fixed, Z be a random variable andt ∈ [s, T ]. Then Es,x[Z|Ft] = Es,x[Z|Fs,x

t ] Ps,x a.s.

Proof. Es,x[Z|Ft] is Ft-measurable and therefore Fs,xt -measurable. Moreover, let

Gs,x ∈ Fs,xt , by Remark 32 in [5] Chapter II, there exists G ∈ Ft such that

Ps,x(G ∪Gs,x) = Ps,x(G\Gs,x) implying 1G = 1Gs,x Ps,x a.s. So

Es,x [1Gs,xEs,x[Z|Ft]] = Es,x [1GEs,x[Z|Ft]]

= Es,x [1GZ]= Es,x [1Gs,xZ] ,

where the second equality occurs because of the definition of Es,x[Z|Ft].

In particular, under the probability measure Ps,x, (Ft)t∈[0,T ]-martingales and(Fs,x

t )t∈[0,T ]-martingales coincide.We now show that in our setup, a canonical Markov class verifies the Blumenthal

0-1 law in the following sense.

Proposition 3.14. Let (s, x) ∈ [0, T ] × E and F ∈ Fs,s. Then Ps,x(F ) is equalto 1 or to 0; In other words, Fs,s is Ps,x-trivial.

Proof. Let F ∈ Fs,s as introduced in Notation 3.1. Since by Remark 3.3, Fs,s =Fs ∩ Fs,T , then F belongs to Fs so by conditioning we get

Es,x[1F ] = Es,x[1F1F ]= Es,x[1FE

s,x[1F |Fs]]= Es,x[1FE

s,Xs [1F ]],


where the latter equality comes from (3.4) because F ∈ Fs,T . But Xs = x, Ps,x

a.s., soEs,x[1F ] = Es,x[1FE

s,x[1F ]]= Es,x[1F ]

2.

3.2. Examples of canonical Markov classes. We will list here some well-known examples of canonical Markov classes and some more recent ones.

• Let E := Rd for some d ∈ N∗. We are given b ∈ Bb(R+ × Rd,Rd),α ∈ Cb(R+×Rd, S∗

+(Rd)) (where S∗

+(Rd) is the space of symmetric strictly

positive definite matrices of size d) and K a Levy kernel (this means thatfor every (t, x) ∈ R+ × Rd, K(t, x, ·) is a σ-finite measure on Rd\0,supt,x

∫ ∥y∥2

1+∥y∥2K(t, x, dy) < ∞ and for every Borel set A ∈ B(Rd\0),

(t, x) 7−→∫A

∥y∥2

1+∥y∥2K(t, x, dy) is Borel) such that for any A ∈ B(Rd\0),(t, x) 7−→

∫A

y1+∥y∥2K(t, x, dy) is bounded continuous.

Let a denote the operator defined on some ϕ ∈ C1,2b (R+ ×Rd) by

∂tϕ+1

2Tr(α∇2ϕ) + (b,∇ϕ) +

∫ (ϕ(·, ·+ y)− ϕ− (y,∇ϕ)

1 + ∥y∥2

)K(·, ·, dy) (3.7)

In [20] (see Theorem 4.3 and the penultimate sentence of its proof), thefollowing is shown. For every (s, x) ∈ R+ × Rd, there exists a uniqueprobability Ps,x on the canonical space (see Definition 3.1) such that

ϕ(·, X·)−∫ ·sa(ϕ)(r,Xr)dr is a local martingale for every ϕ ∈ C1,2

b (R+×Rd)and Ps,x(Xs = x) = 1. Moreover (Ps,x)(s,x)∈R+×Rd defines a canonicalMarkov class and its transition kernel is measurable in time.

• The case K = 0 was studied extensively in the celebrated book [21] inwhich it is also shown that if b, α are bounded and continuous in the secondvariable, then there exists a canonical Markov class with transition kernelmeasurable in time (Ps,x)(s,x)∈R+×Rd such that ϕ(·, X·)−

∫ ·sa(ϕ)(r,Xr)dr

is a local martingale for any ϕ ∈ C1,2b (R+ ×Rd).

• In [19], a canonical Markov class whose transition kernel is the weak fun-damental solution of a parabolic PDE in divergence form is exhibited.

• In [13], diffusions on manifolds are studied and shown to define canonicalMarkov classes.

• Solutions of PDEs with distributional drift are exhibited in [12] and shownto define canonical Markov classes.

Some of previous examples were only studied as homogeneous Markov processesbut can easily be shown to fall in the non-homogeneous setup of the present paperas it was illustrated in [3].

4. Martingale Additive Functionals

We now introduce the notion of non-homogeneous Additive Functional that weuse in the paper. This looks to be a good compromise between the notion of Addi-tive Functional associated to a stochastic system introduced by E.B. Dynkin (see


for example [9]) and the more popular notion of homogeneous Additive Functionalstudied extensively, for instance by C. Dellacherie and P.A. Meyer in [7] ChapterXV. This section consists in extending some essential results stated in [7] ChapterXV to our setup.

Our framework is still the canonical space introduced at Notation 3.1. In par-ticular X is the canonical process.

Definition 4.1. We denote ∆ := (t, u) ∈ [0, T ]2|t ≤ u. On (Ω,F), we definea non-homogeneous Additive Functional (shortened AF) as a random-field A :=(Atu)(t,u)∈∆ indexed by ∆ with values in R, verifying the two following conditions.

(1) For any (t, u) ∈ ∆, Atu is Ft,u-measurable;(2) for any (s, x) ∈ [0, T ]×E, there exists a real cadlag Fs,x-adapted process

As,x (taken equal to zero on [0, s] by convention) such that for any x ∈ Eand s ≤ t ≤ u, Atu = As,xu −As,xt Ps,x a.s.

As,x will be called the cadlag version of A under Ps,x.An AF will be called a non-homogeneous square integrable Martingale Additive

Functional (shortened square integrable MAF) if under any Ps,x its cadlag versionis a square integrable martingale. More generally an AF will be said to verify acertain property (being non-negative, increasing, of bounded variation, squareintegrable, having L1-terminal value) if under any Ps,x its cadlag version verifiesit.

Finally, given an increasing AF A and an increasing function V , A will besaid to be absolutely continuous with respect to V if for any (s, x) ∈ [0, T ] × E,dAs,x ≪ dV in the sense of stochastic measures.

Remark 4.2. Let (Px)x∈E be a homogeneous canonical Markov class (see Notation3.9). We recall that in the classical literature (see Definition 3 of [7] for instance),an adapted right-continuous process A on the canonical space is called an AdditiveFunctional if for all 0 ≤ t ≤ u ≤ T and x ∈ E

Au = At +Au−t θt Px a.s., (4.1)

where θt : ω 7→ ω ((t+ ·) ∧ T ) denotes the shift operator at time t.Let (Ps,x)(s,x)∈[0,T ]×E be the canonical Markov class related to (Px)x∈E in the

sense of Notation 3.9. If for every 0 ≤ t ≤ u ≤ T , Equation (4.1) holds for all ω,then the random field (t, u) 7−→ Au−At is a non-homogeneous Additive Functionalin the sense of Definition 4.1.

Example 4.3. Let ϕ ∈ C([0, T ]×E,R), ψ ∈ Bb([0, T ]×E,R) and V : [0, T ] 7−→ Rbe right-continuous and non-decreasing function. Then the random field A givenby

Atu := ϕ(u,Xu)− ϕ(t,Xt)−∫ u

t

ψ(r,Xr)dVr, (4.2)

defines a non-homogeneous Additive Functional. Its cadlag version under Ps,x

may be given by

As,x = ϕ(· ∨ s,X·∨s)− ϕ(s, x)−∫ ·∨s

s

ψ(r,Xr)dVr. (4.3)


We now adopt the setup of the first item of Section 3.2. We consider some ϕ ∈C1,2b ([0, T ]× Rd), then the random field M given by

M tu := ϕ(u,Xu)− ϕ(t,Xt)−

∫ u

t

a(ϕ)(r,Xr)dr, (4.4)

defines a square integrable MAF with cadlag version under Ps,x given by

Ms,x = ϕ(· ∨ s,X·∨s)− ϕ(s, x)−∫ ·∨s

s

a(ϕ)(r,Xr)dr. (4.5)

In this section for a given MAF (M tu)(t,u)∈∆ we will be able to exhibit two AF,

denoted respectively by ([M ]tu)(t,u)∈∆ and (⟨M⟩tu)(t,u)∈∆, which will play respec-tively the role of a quadratic variation and an angular bracket of it. Moreover wewill show that the Radon-Nikodym derivative of the mentioned angular bracket ofa MAF with respect to our reference function V is a time-dependent function ofthe underlying process.

Proposition 4.4. Let (M tu)(t,u)∈∆ be a square integrable MAF, and for any

(s, x) ∈ [0, T ] × E, [Ms,x] be the quadratic variation of its cadlag version Ms,x

under Ps,x. Then there exists an AF which we will call ([M ]tu)(t,u)∈∆ and which,for any (s, x) ∈ [0, T ]× E, has [Ms,x] as cadlag version under Ps,x.

Proof. We adapt Theorem 16 Chapter XV in [7] to a non homogeneous set-up butthe reader must keep in mind that our definition of Additive Functional is differentfrom the one related to the homogeneous case.

For the whole proof t < u will be fixed. We consider a sequence of subdivisionsof [t, u]: t = tk1 < tk2 < · · · < tkk = u such that min

i<k(tki+1 − tki ) −→

k→∞0. Let (s, x) ∈

[0, t]×E with corresponding probability Ps,x. For any k, we have∑i<k

(M

tkitki+1

)2=∑

i<k

(Ms,x

tki+1

−Ms,x

tki)2 Ps,x a.s., so by definition of quadratic variation we know that

∑i<k

(M

tkitki+1

)2Ps,x

−→k→∞

[Ms,x]u − [Ms,x]t. (4.6)

In the sequel we will construct an Ft,u-measurable random variable [M ]tu such that

for any (s, x) ∈ [0, t]× E,∑i≤k

(M

tkitki+1

)2Ps,x

−→k→∞

[M ]tu. In that case [M ]tu will then

be Ps,x a.s. equal to [Ms,x]u − [Ms,x]t.

Let x ∈ E. Since M is a MAF, for any k,∑i<k

(M

tkitki+1

)2is Ft,u-measurable and

therefore F t,xt,u -measurable. Since F t,x

t,u is complete, the limit in probability of this

sequence, [M t,x]u − [M t,x]t, is still F t,xt,u -measurable. By Proposition 3.12, there

is an Ft,u-measurable variable which depends on (t, x), that we call at(x, ω) suchthat

at(x, ω) = [M t,x]u − [M t,x]t,Pt,x a.s. (4.7)

We will show below that there is a jointly measurable version of (x, ω) 7→ at(x, ω).For every integer n ≥ 0, we set ant (x, ω) := n∧at(x, ω) which is in particular limit


in probability of n∧∑i≤k

(M

tkitki+1

)2under Pt,x. For any integers k, n and any x ∈ E,

we define the finite positive measures Qk,n,x, Qn,x and Qx on (Ω,Ft,u) by

(1) Qk,n,x(F ) := Et,x[1F

(n ∧

∑i<k

(M

tkitki+1

)2)];

(2) Qn,x(F ) := Et,x[1F (ant (x, ω))];(3) Qx(F ) := Et,x[1F (at(x, ω))].

When k and n are fixed, for any fixed F , by Proposition 3.10, the mapping

x 7−→ Et,x[F

(n ∧

∑i<k

(M

tkitki+1

)2)]is Borel.

Then n ∧∑i<k

(M

tkitki+1

)2Pt,x

−→k→∞

ant (x, ω), and this sequence is uniformly bounded

by the constant n, so the convergence takes place in L1, therefore x 7−→ Qn,x(F )is also Borel as the pointwise limit in k of the functions x 7−→ Qk,n,x(F ). Sim-

ilarly, ant (x, ω)a.s.−→n→∞

at(x, ω) and is non-decreasing, so by monotone convergence

theorem, being a pointwise limit in n of the functions x 7−→ Qn,x(F ), the functionx 7−→ Qx(F ) is Borel. We recall that F is separable. The just two mentionedproperties and the fact that, for any x, we also have (by item 3. above)Qx ≪ Pt,x,allows to show (see Theorem 58 Chapter V in [6]) the existence of a jointly mea-surable (for B(E)⊗Ft,u) version of (x, ω) 7→ at(x, ω), that we recall to be densitiesof Qx with respect to Pt,x. That version will still be denoted by the same symbol.

We can now set [M ]tu(ω) = at(Xt(ω), ω), which is a correctly defined Ft,u-measurable random variable. For any x, since Pt,x(Xt = x) = 1, we have theequalities

[M ]tu = at(x, ·) = [M t,x]u − [M t,x]t Pt,xa.s. (4.8)

We will moreover prove that

[M ]tu = [Ms,x]u − [Ms,x]t Ps,x a.s., (4.9)

holds for every (s, x) ∈ [0, t]× E, and not just in the case s = t that we have justestablished in (4.8).

Let us fix s < t and x ∈ E. We show that under any Ps,x, [M ]tu is the limit in

probability of∑i<k

(M

tkitki+1

)2. Indeed, let ϵ > 0: the event∣∣∣∣∣∑i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣∣ > ϵ

belongs to Ft,T so by conditioning and using the Markov property (3.4) we have

Ps,x(∣∣∣∣∑

i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣ > ϵ

)= Es,x

[Ps,x

(∣∣∣∣∑i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣ > ϵ

∣∣∣∣Ft)]= Es,x

[Pt,Xt

(∣∣∣∣∑i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣ > ϵ

)].


For any fixed y, by (4.6) and (4.8), Pt,y(∣∣∣∣∑

i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣ > ϵ

)tends to

zero when k goes to infinity, for every realization ω, it yields that

Pt,Xt

(∣∣∣∣∣∑i<k

(M

tkitki+1

)2− [M ]tu

∣∣∣∣∣ > ϵ

)tends to zero when k goes to infinity. Since this sequence is dominated by theconstant 1, that convergence still holds under the expectation with respect to theprobability the probability Ps,x, thanks to the dominated convergence theorem.

So we have built an Ft,u-measurable variable [M ]tu such that under any Ps,x

with s ≤ t, [Ms,x]u − [Ms,x]t = [M ]tu a.s. and this concludes the proof.

We will now extend the result about quadratic variation to the angular bracketof MAFs. The next result can be seen as an extension of Theorem 15 Chapter XVin [7] to a non-homogeneous context.

Proposition 4.5. Let (Btu)(t,u)∈∆ be an increasing AF with L1-terminal value,for any (s, x) ∈ [0, T ]×E, let Bs,x be its cadlag version under Ps,x and let As,x bethe predictable dual projection of Bs,x in (Ω,Fs,x, (Fs,x

t )t∈[0,T ],Ps,x). Then there

exists an increasing AF with L1 terminal value (Atu)(t,u)∈∆ such that under anyPs,x, the cadlag version of A is As,x.

Proof. The first half of the demonstration will consist in showing that

∀(s, x) ∈ [0, t]× E, (As,xu −As,xt ) is Fs,xt,u−measurable. (4.10)

We start by recalling a property of the predictable dual projection which we willhave to extend slightly. Let us fix (s, x) and the corresponding stochastic basis(Ω,Fs,x, (Fs,x

t )t∈[0,T ],Ps,x). For any F ∈ Fs,x, let Ns,x,F be the cadlag version of

the martingale, r 7−→ Es,x[1F |Fr]. Then for any 0 ≤ t ≤ u ≤ T , the predictable

projection of the process r 7→ 1F1[t,u[(r) is r 7→ Ns,x,Fr− 1[t,u[(r), see the proof of

Theorem 43 Chapter VI in [6]. Therefore by definition of the dual predictableprojection (see Definition 73 Chapter VI in [6]) we have

Es,x [1F (As,xu −As,xt )] = Es,x

[∫ u

t

Ns,x,Fr− dBs,xr

], (4.11)

for any F ∈ Fs,x.We will now prove some technical lemmas which in a sense extend this property,

and will permit us to operate with a good common version of the random variable∫ utNs,x,Fr− dBs,xr not depending on (s, x).

For the rest of the proof, 0 ≤ t < u ≤ T will be fixed.

Notation 4.6. Let F ∈ Ft,T . We denote for any r ∈ [t, T ], ω ∈ Ω, NFr (ω) :=

Pt,Xt(ω)(F ).

It is clear that NF previously introduced is an (Ft,r)r∈[t,T ]-adapted processwhich does not depend on (s, x), which takes values in [0, 1] for all r, ω and byProposition 3.8, for any (s, x) ∈ [0, t] × E, Ns,x,F is, on [t, T ], a Ps,x-version ofNF .


Lemma 4.7. Let F ∈ Ft,T . There exists an Ft,u-measurable random variablewhich we will denote

∫ utNFr−dBr such that for any (s, x) ∈ [0, t]×E,

∫ utNFr−dBr =∫ u

tNs,x,Fr− dBs,xr Ps,x a.s.

Remark 4.8. By definition, the process NF introduced in Notation 4.6 and ther.v.

∫ utNFr−dBr will not depend on any (s, x).

Proof. In some sense we wish to integrate r 7→ NFr− against Bt for fixed ω. How-

ever first we do not know a priori if the paths r 7→ NFr and r 7→ Btr are measurable,

second r 7→ NFr may not have a left limit and Bt may be not of bounded vari-

ation. So it is not clear if∫ utNFr−dB

tr makes sense for any ω. Moreover under

a certain Ps,x, NF,s,x and Bs,x· − Bs,xt are only versions of NF and Bt and notindistinguishable to them. Even if we could compute the aforementioned integral,

it would not be clear if∫ utNFr−dB

tr =

∫ utNs,x,Fr− dBs,xr Ps,x a.s.

We start by some considerations about B, settingWtu := ω : supr∈[t,u]∩Q

Btr <∞

which is Ft,u-measurable, and for r ∈ [t, u]

Btr(ω) :=

supt≤v<rv∈Q

Btv(ω) if ω ∈Wtu

0 otherwise.

Bt is an increasing, finite (for all ω) process. In general, it is neither a measurablenor an adapted process; however for any r ∈ [t, u], Btr is still Ft,u-measurable.Since it is increasing, it has right and left limits at each point for every ω, so wecan define the process Bt indexed on [t, u] below:

Btr := limv↓rv∈Q

Btv, r ∈ [t, u], (4.12)

when u ∈]t, T [ and BtT := BtT if u = T . Therefore Bt is an increasing, cadlagprocess. It is constituted by Ft,u-measurable random variables, and by Theorem

15 Chapter IV of [5], Bt is a also a measurable process (indexed by [t, u]).

We can show that Bt is Ps,x-indistinguishable from Bs,x· −Bs,xt for any (s, x) ∈[0, t] × E. Indeed, let (s, x) be fixed. Since Bs,x· − Bs,xt is a version of Bt andQ being countable, there exists a Ps,x-null set N such that for all ω ∈ N c andr ∈ Q∩ [t, u], Bs,xr (ω)−Bs,xt (ω) = Btr(ω). Therefore for any ω ∈ N c and r ∈ [t, u],

Btr(ω) = limv↓rv∈Q

supt≤w<vw∈Q

Btw(ω) = limv↓rv∈Q

supt≤w<vw∈Q

Bs,x(ω)w −Bs,x(ω)t

= Bs,x(ω)r −Bs,x(ω)t,

where the latter equality comes from the fact that Bs,x(ω) is cadlag and increasing.So we have constructed an increasing finite cadlag (for all ω) process and so the

path r 7→ Bt(ω) is a Lebesgue integrator on [t, u] for each ω.We fix now F ∈ Ft,T and we discuss some issues related to NF . Since it

is positive, we can start defining the process N , for index values r ∈ [t, T [ byNFr := liminf

v↓rv∈Q

NFv , and setting NF

T := NFT . This process is (by similar arguments as


for Bt defined in (4.12)), Ps,x-indistinguishable to Ns,x,F for all (s, x) ∈ [0, t]×E.For any r ∈ [t, T ], NF

r (see Notation 4.6) is Ft,r-measurable, so NFr will also be

Ft,r-measurable for any r ∈ [t, T ] by right-continuity of Ft,· (see Notation 3.1).However, NF is not necessarily cadlag for every ω, and also not necessarily ameasurable process. We subsequently define

W ′tu := ω ∈ Ω|there is a cadlag function f such that NF (ω) = f on [t, u] ∩Q.

By Theorem 18 b) in Chapter IV of [5], W ′tu is Ft,u-measurable so we can define

on [t, u] NFr := NF

r 1W ′tu. NF is no longer (Ft)t∈[0,T ]-adapted, however, it is now

cadlag for all ω and therefore a measurable process by Theorem 15 Chapter IV of[5]. The r.v. NF

r are still Ft,u-measurable , and NF is still Ps,x-indistinguishableto Ns,x,F on [t, u] for any (s, x) ∈ [0, t]× E.

Finally we can define∫ utNFr−dBr :=

∫ utNFr−dB

tr which is Ps,x a.s. equal to∫ u

tNs,x,Fr− dBs,xr for any (s, x) ∈ [0, t] × E. Moreover, since NF and B are both

measurable with respect to B([t, u])⊗Ft,u, then∫ utNFr−dBr is Ft,u-measurable.

The lemma below is a conditional version of the property (4.11).

Lemma 4.9. For any (s, x) ∈ [0, t]× E and F ∈ Fs,xt,T we have Ps,x-a.s.

Es,x [1F (As,xu −As,xt )|Ft] = Es,x

[∫ u

t

NFr−dBr

∣∣∣∣Ft] .Proof. Let s, x, F be fixed. By definition of conditional expectation, we need toshow that for any G ∈ Ft we have

Es,x [1G1F (As,xu −As,xt )] = Es,x

[1GE

s,x

[∫ u

t

NFr−dBr

∣∣∣∣Ft]] a.s.

For r ∈ [t, u] we have Es,x[1F∩G|Fr] = 1GEs,x[1F |Fr] a.s. therefore the cadlag

versions of those processes are indistinguishable on [t, u] and the random variables∫ utNG∩Fr− dBr and 1G

∫ utNFr−dBr as defined in Lemma 4.7 are a.s. equal. So by

the non conditional property of dual predictable projection (4.11) we have

Es,x [1G1F (As,xu −As,xt )] = Es,x

[∫ utNG∩Fr− dBr

]= Es,x

[1G∫ utNFr−dBr

]= Es,x

[1GE

s,x[∫ utNFr−dBr

∣∣Ft]] ,which concludes the proof.

Lemma 4.10. For any (s, x) ∈ [0, t]× E and F ∈ Ft,T we have Ps,x-a.s.,

Es,x [1F (As,xu −As,xt )|Ft] = Es,x [1F (As,xu −As,xt )|Xt] .

Proof. By Lemma 4.9 we have

Es,x [1F (As,xu −As,xt )|Ft] = Es,x

[∫ u

t

NFr−dBr

∣∣∣∣Ft] .By Lemma 4.7,

∫ utNFr−dBr is Ft,T measurable so the Markov property (3.4) im-

plies

Es,x[∫ u

t

NFr−dBr

∣∣∣∣Ft] = Es,x [∫ u

t

NFr−dBr

∣∣∣∣Xt

],


therefore Es,x [1F (As,xu −As,xt )|Ft] is a.s. equal to a σ(Xt)-measurable r.v and so

is a.s. equal to Es,x [1F (As,xu −As,xt )|Xt] .

We are now able to prove (4.10) which is the first important issue of the proofof Proposition 4.5, which states that By definition, a predictable dual projectionis adapted so we already know that (As,xu −As,xt ) is Fs,x

u -measurable, therefore byRemark 3.3, it is enough to show that it is also Fs,x

t,T -measurable. So we are goingto show that

As,xu −As,xt = Es,x [As,xu −As,xt |Ft,T ] Ps,x a.s. (4.13)

For this we will show that

Es,x [1F (As,xu −As,xt )] = Es,x [1FE

s,x [As,xu −As,xt |Ft,T ]] , (4.14)

for any F ∈ F . We will prove (4.14) for F ∈ F event of the form F = Ft∩Ft,T withFt ∈ Ft and Ft,T ∈ Ft,T . By item 4. of Remark 3.3, such events form a π-systemΠ which generates F . Consequently, by the monotone class theorem, (4.14) willremain true for any F ∈ F and even in Fs,x since Ps,x-null set will not impact theequality. This will imply (4.13) so that As,xu − As,xt is Fs,x

t,T -measurable. At this

point, as we have anticipated, we prove (4.14) for a fixed F = Ft ∩ Ft,T ∈ Π. ByLemma 4.10 we have


[1FtE

s,x[1Ft,T (A

s,xu −As,xt )|Ft

]]= Es,x

[1FtE

s,x[1Ft,T (A

s,xu −As,xt )|Xt

]]= Es,x

[1FtE

s,x[Es,x

[1Ft,T (A

s,xu −As,xt )|Ft,T

]|Xt

]],

where the latter equality holds since σ(Xt) ⊂ Ft,T . Now observe that sinceEs,x

[1Ft,T (A


]is Ft,T -measurable, the Markov property (3.4) al-

lows us to substitute the conditional σ-field σ(Xt) with Ft and obtain


[1FtE

s,x[Es,x

[1Ft,T (A


]|Ft]]

= Es,x[1FtE

s,x[1Ft,T (A


]]= Es,x

[1Ft1Ft,TE

s,x [(As,xu −As,xt )|Ft,T ]]

= Es,x [1FEs,x [(As,xu −As,xt )|Ft,T ]] .

This concludes the proof of (4.14), therefore (4.13) holds so that As,xu − As,xt isFs,xt,u -measurable and so (4.10) is established. This concludes the first part of the

proof of Proposition 4.5.We pass to the second part of the proof of Proposition 4.5 where we will show

that for given 0 < t < u there is an Ft,u-measurable r.v. Atu such that for every(s, x) ∈ [0, t]× E, (As,xu −As,xt ) = Atu Ps,x a.s.

Similarly to what we did with the quadratic variation in Proposition 4.4, westart by noticing that for any x ∈ E, since (At,xu −At,xt ) is F t,x

t,u -measurable, thereexists by Proposition 3.12 an Ft,u-measurable r.v. a(x, ω) such that

a(x, ω) = At,xu −At,xt Pt,x a.s. (4.15)

As in the proof of Proposition 4.4, we will show the existence of a jointly measur-able version of (x, ω) 7→ a(x, ω). For every x ∈ E we define on Ft,u the positivemeasure

Qx : F 7−→ Et,x[1F (A

t,xu −At,xt )

]= Et,x [1Fa(x, ω)] . (4.16)


By Lemma 4.7, and (4.11), for every F ∈ Ft,u we have

Qx(F ) = Et,x[∫ u

t

NFr−dBr

], (4.17)

and we recall that∫ utNFr−dBr does not depend on x. So by Proposition 3.10

x 7−→ Qx(F ) is Borel for any F . Moreover, for any x, Qx ≪ Pt,x. Again byTheorem 58 Chapter V in [6], there exists a version (x, ω) 7→ a(x, ω) measurablefor B(E)⊗Ft,u of the related Radon-Nikodym densities.

We can now set Atu(ω) := a(Xt(ω), ω) which is then an Ft,u-measurable r.v.Since Pt,x(Xt = x) = 1 and (4.15) hold, we have

Atu = a(Xt, ·) = a(x, ·) = At,xu −At,xt Pt,x a.s. (4.18)

We now fix s < t and x ∈ E and we want to show that we still haveAtu = As,xu −As,xtPs,x a.s. So, as above, we consider F ∈ Ft,u and, thanks to (4.11) we compute


[∫ utNFr−dBr

]= Es,x

[Es,x

[∫ utNFr−dBr|Ft

]]= Es,x

[Et,Xt

[∫ utNFr−dBr

]]= Es,x

[Et,Xt [1FA

tu]]

= Es,x [Es,x [1FAtu|Ft]]

= Es,x [1FAtu] .

(4.19)

Indeed, concerning the fourth equality we recall that, by (4.16), (4.17) and (4.18),we have Et,x

[∫ utNFr−dBr

]= Et,x [1FA

tu] for all x, so this equality becomes an

equality whatever random variable we plug into x. The third and fifth equal-ities come from the Markov property (3.4) since

∫ utNFr−dBr and Atu are Ft,T -

measurable. Then, adding Ps,x-null sets does not change the validity of (4.19), sowe have for any F ∈ Fs,x

t,u that Es,x [1F (As,xu −As,xt )] = Es,x [1FA

tu].

Finally, since we had shown in the first half of the proof that As,xu − As,xt isFs,xt,u -measurable, and since Atu also has, by construction, the same measurability

property, we can conclude that As,xu −As,xt = Atu Ps,x a.s.

Since this holds for every t ≤ u and (s, x) ∈ [0, t]×E, (Atu)(t,u)∈∆ is the desiredAF, which ends the proof of Proposition 4.5.

Corollary 4.11. LetM , M ′ be two square integrable MAFs, letMs,x (respectivelyM ′s,x) be the cadlag version of M (respectively M ′) under Ps,x. Then there existsa bounded variation AF with L1 terminal condition denoted ⟨M,M ′⟩ such thatunder any Ps,x, the cadlag version of ⟨M,M ′⟩ is ⟨Ms,x,M ′s,x⟩. If M = M ′ theAF ⟨M,M ′⟩ will be denoted ⟨M⟩ and is increasing.

Proof. If M = M ′, the corollary comes from the combination of Propositions 4.4and 4.5, and the fact that the angular bracket of a square integrable martingaleis the dual predictable projection of its quadratic variation. Otherwise, it is clearthat M +M ′ and M −M ′ are square integrable MAFs, so we can consider theincreasing MAFs ⟨M −M ′⟩ and ⟨M +M ′⟩. We introduce the AF

⟨M,M ′⟩ = 1

4(⟨M +M ′⟩ − ⟨M −M ′⟩),


which by polarization has cadlag version ⟨Ms,x,M ′s,x⟩ under Ps,x. ⟨M,M ′⟩ istherefore a bounded variation AF with L1 terminal condition.

We are now going to study the Radon-Nikodym derivative of an increasingcontinuous AF with respect to some measure. The next result can be seen as anextension of Theorem 13 Chapter XV in [7] in a non-homogeneous setup. We willneed the following lemma.

Lemma 4.12. Let (E, E) be a measurable space, let I be a sub-interval of R+ andlet f : E×I −→ R be a mapping such that for all t ∈ I, x 7→ f(x, t) is measurablewith respect to E and for all x ∈ E, t 7→ f(x, t) is right-continuous, then f ismeasurable with respect to E ⊗ B(I).

Proof. On (E, E) we introduce the filtration (Et)t∈I where Et = E for all t. In thefiltered space (E, E , (Et)t∈I), f defines a right-continuous adapted process and istherefore progressively measurable (see Theorem 15 in [5] Chapter IV for instance),and in particular it is measurable. This means that f is measurable with respectto E ⊗ B(I).

Proposition 4.13. Let A be a positive, non-decreasing AF absolutely continuouswith respect to some continuous non-decreasing function V , and for every (s, x) ∈[0, T [×E let As,x be the cadlag version of A under Ps,x. There exists a Borelfunction h ∈ B([0, T ] × E,R) such that for every (s, x) ∈ [0, T ] × E, As,x =∫ ·∨ss

h(r,Xr)dVr, in the sense of indistinguishability.

Proof. We set

Ctu = Atu + (Vu − Vt) + (u− t), (4.20)

which is an AF with cadlag versions

Cs,xt = As,xt + Vt + t, (4.21)

and we start by showing the statement for A and C instead of A and V . We

introduce the intermediary function C so that for any u > t thatAs,x

u −As,xt

Cs,xu −Cs,x

t∈

[0, 1]; that property will be used extensively in connections with the applicationof dominated convergence theorem.

Since As,x is non-decreasing for any (s, x) ∈ [0, T ]×E, A can be taken positive(in the sense that Atu(ω) ≥ 0 for any (t, u) ∈ ∆ and ω ∈ Ω) by considering A+

(defined by (A+)tu(ω) := Atu(ω)+) instead of A.

For t ∈ [0, T [ we set

Kt := liminfn→∞

Att+ 1

n

Att+ 1

n

+ 1n + (Vt+ 1

n− Vt)

= limn→∞

infp≥n

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

(4.22)

= limn→∞

limm→∞

minn≤p≤m

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

.


By positivity, this liminf always exists and belongs to [0, 1] since the sequencebelongs to [0, 1]. For every (s, x) ∈ [0, T ]×E, since for all t ≥ s and n ≥ 0, At

t+ 1n

=

As,xt+ 1

n

− As,xt Ps,x a.s., then Ks,x defined by Ks,xt := liminf

n→∞

As,x

t+ 1n

−As,xt

Cs,x

t+ 1n

−Cs,xt

is a Ps,x-

version of K, for t ∈ [s, T [. By Lebesgue Differentiation theorem (see Theorem 12Chapter XV in [7] for a version of the theorem with a general atomless measure),for any (s, x), for Ps,x-almost all ω, since dCs,x(ω) is absolutely continuous withrespect to dAs,x(ω), Ks,x(ω) is a density of dAs,x(ω) with respect to dCs,x(ω).

We now show that there exists a Borel function k in B([0, T [×E,R) such thatunder any Ps,x, k(t,Xt) is on [s, T [ a version of K (and therefore of Ks,x). Forevery t ∈ [0, T [, Kt is measurable with respect to

∩n≥0

Ft,t+ 1n= Ft,t by construction,

taking into account Notation 3.1. So for any (t, x) ∈ [0, T ] × E, by Proposition3.14, there exists a constant which we denote k(t, x) such that

Kt = k(t, x), Pt,xa.s. (4.23)

For any integers (n,m), we define

kn,m : (t, x) 7→ Et,x

minn≤p≤m

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

,and for any n

kn : (t, x) 7→ Et,x

infp≥n

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

, (4.24)

We start showing that

kn,m :(s, x, t) 7−→ Es,x

[minn≤p≤m

At

t+ 1p

At

t+ 1p

+ 1p+(V

t+ 1p−Vt)

]1s≤t,

[0, T ]× E × [0, T [ −→ [0, 1],

(4.25)

is jointly Borel. In order to do so, we will show that at fixed t, kn,m(·, ·, t) is Borel,at fixed (s, x), kn,m(s, x, ·) is right-continuous and we will conclude on the jointmeasurability thanks to Lemma 4.12.

If we fix t ∈ [0, T [, then by Proposition 3.10

(s, x) 7−→ Es,x

minn≤p≤m

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

is a Borel map. Since (s, x) 7→ 1[t,T ](s) is obviously Borel, considering the

product of the two previous maps, kn,m(·, ·, t) is Borel. We now fix some (s, x)

and show that kn,m(s, x, ·) is right-continuous. Since that function is equal tozero on [0, s[, showing its continuity on [s, T [ will be sufficient. We remarkthat As,x is continuous Ps,x a.s. V is continuous, and the minimum of a fi-nite number of continuous functions remains continuous. Let tq −→

q→∞t be a


converging sequence in [s, T [. Then minn≤p≤m

As,x

tq+ 1p

−As,xtq

As,x

tq+ 1p

−As,xtq

+ 1p+(V

tq+ 1p−Vtq )

tends a.s.

to minn≤p≤m

As,x

t+ 1p

−As,xt

As,x

t+ 1p

−As,xt + 1

p+(Vt+ 1

p−Vt)

, when q tends to infinity. Since for any s ≤

t ≤ u, Atu = As,xu − As,xt Ps,x a.s., thenA

tq

tq+ 1p

Atq

tq+ 1p

+ 1p+(V

tq+ 1p−Vtq )

tends a.s. to

At

t+ 1p

At

t+ 1p

+ 1p+(V

t+ 1p−Vt)

. All those terms belonging to [0, 1], by dominated convergence

theorem, the mentioned convergence also holds under the expectation, hence theannounced continuity related to kn,m is established and as anticipated, kn,m isjointly measurable in all its variables.

Since kn,m(t, y) = kn,m(t, t, y), by composition we can deduce that for any n,m,kn,m is Borel. By the dominated convergence theorem, kn,m tends pointwise tokn (which was defined in (4.24), when m goes to infinity so kn are also Borel forevery n. Finally, keeping in mind (4.22) nd (4.23) we have Pt,x a.s.

k(t, x) = Kt = limn→∞

infp≥n

Att+ 1

p

Att+ 1

p

+ 1p + (Vt+ 1

p− Vt)

.

Taking the expectation and again by the dominated convergence theorem, kn

(defined in (4.24)) tends pointwise to k when n goes to infinity so k is Borel.We now show that, for any (s, x) ∈ [0, T ] × E, k(·, X·) is a Ps,x-version of K

on [s, T [. Since Pt,x(Xt = x) = 1, we know that for any t ∈ [0, T ], x ∈ E, wehave Kt = k(t, x) = k(t,Xt) P

t,x-a.s., and we prove below that for any t ∈ [0, T ],(s, x) ∈ [0, t]× E, we have Kt = k(t,Xt) P

s,x-a.s.

Let t ∈ [0, T ] be fixed. Since A is an AF, for any n,At

t+ 1p

At

t+ 1p

+ 1n+(V

t+ 1n−Vt)

is

Ft,t+ 1n-measurable. So the event

liminfn→∞

Att+ 1

n

Att+ 1

n

+ 1n + (Vt+ 1

n− Vt)

= k(t,Xt)

belongs to Ft,T and by Markov property (3.4), for any (s, x) ∈ [0, t]× E, we get

Ps,x(Kt = k(t,Xt)) = Es,x[Ps,x (Kt = k(t,Xt)|Ft)]= Es,x[Pt,Xt (Kt = k(t,Xt))]

= 1.

For any (s, x), the process k(·, X·) is therefore on [s, T [ a Ps,x-modification ofK and therefore of Ks,x. However it is not yet clear if provides another density ofdAs,x with respect to dCs,x, which was defined at (4.21).

Considering that (t, u, ω) 7→ Vu − Vt also defines a positive non-decreasing AFabsolutely continuous with respect to C, defined in (4.20), we proceed similarly asat the beginning of the proof, replacing the AF A with V .


Let the process K ′ be defined by

K ′t = liminf

n→∞

Vt+ 1n− Vt

Att+ 1

n

+ 1n + (Vt+ 1

n− Vt)

,

and for any(s, x), let K ′s,x be defined on [s, T [ by

K ′s,xt = liminf

n→∞

Vt+ 1n− Vt

As,xt+ 1

n

−As,xt + 1n + (Vt+ 1

n− Vt)

.

Then, for any (s, x), K ′s,x on [s, T [ is a Ps,x-version of K ′, and it constitutes adensity of dV (ω) with respect to dCs,x(ω) on [s, T [, for almost all ω. One showsthen the existence of a Borel function k′ such that for any (s, x), k′(·, X·) is aPs,x-version of K ′ and a modification of K ′s,x on [s, T [. So for any (s, x), underPs,x, we can write

As,x =∫ ·∨ss

Ks,xr dCs,xr

V·∨s − Vs =∫ ·∨ss

K ′s,xr dCs,xr

Now since dAs,x ≪ dV , for a fixed ω, the set r ∈ [s, T ]|K ′s,xr (ω) = 0 is negligible

with respect to dV so also for dAs,x(ω) and therefore we can write

As,x =∫ ·∨ss

Ks,xr dCs,xr

=∫ ·∨ss

Ks,xr

K′s,xr

1K′s,xr =0K

′s,xr dCs,xr

+∫ ·∨ss

1K′s,xr =0dA

s,xr

=∫ ·∨ss

Ks,xr

K′s,xr

1K′s,xr =0dVr,

where we use the convention that for any two functions ϕ, ψ then ϕψ1ψ =0 is defined

by by

ϕ

ψ1ψ =0(x) =

ϕ(x)ψ(x) if ψ(x) = 0

0 if ψ(x) = 0.

We set now h := kk′1k′r =0 which is Borel, and clearly for any (s, x), h(t,Xt)

is a Ps,x-version of Hs,x := Ks,x

K′s,x1K′s,x =0 on [s, T [. So by Lemma 5.12 in [2],Hs,xt = h(t,Xt) dV ⊗ dPs,x a.e. and finally we have shown that under any Ps,x,

As,x =∫ ·∨ss

h(r,Xr)dVr on [0, T [. Without change of notations we extend h to[0, T ] × E by zero for t = T . Since As,x is continuous Ps,x-a.s. previous equalityextends to T . Proposition 4.14. Let (Atu)(t,u)∈∆ be an AF with bounded variation and taking

L1 values. Then there exists an increasing AF which we denote (Pos(A)tu)(t,u)∈∆

(resp. (Neg(A)tu)(t,u)∈∆ ) and which, for any (s, x) ∈ [0, T ] × E, has Pos(As,x)(resp. Neg(As,x)) as cadlag version under Ps,x.

Proof. By definition of the total variation of a bounded variation function, thefollowing holds. For every (s, x) ∈ [0, T ] × E, s ≤ t ≤ u ≤ T for Ps,x almost allω ∈ Ω, and any sequence of subdivisions of [t, u]: t = tk1 < tk2 < · · · < tkk = u suchthat min


k→∞0 we have∑

i<k

|As,xtki+1

(ω)−As,xtki

(ω)| −→k→∞

V ar(As,x)u(ω)− V ar(As,x)t(ω), (4.26)


taking into account the considerations of the end of Section 2. By Proposition3.3 in [15] Chapter I, we have Pos(As,x) = 1

2 (V ar(As,x) +As,x) and Neg(As,x) =

12 (V ar(A

s,x)−As,x). Moreover, for any x ∈ R we know that x+ = 12 (|x|+ x) and

x− = 12 (|x| − x), so we also have∑i<k

(As,xtki+1

(ω)−As,xtki

(ω))+ −→k→∞

Pos(As,x)u(ω)− Pos(As,x)t(ω)∑i<k

(As,xtki+1

(ω)−As,xtki

(ω))− −→k→∞

Neg(As,x)u(ω)−Neg(As,x)t(ω),(4.27)

for Ps,x almost all ω. Since the convergence a.s. implies the convergence inprobability, for every (s, x) ∈ [0, T ]×E, s ≤ t ≤ u and any sequence of subdivisionsof [t, u]: t = tk1 < tk2 < · · · < tkk = u such that min


k→∞0, we have

∑i<k

(Atkitki+1

)+Ps,x

−→k→∞

Pos(As,x)u − Pos(As,x)t∑i<k

(Atkitki+1

)−Ps,x

−→k→∞

Neg(As,x)u −Neg(As,x)t.(4.28)

The proof can now be performed according to the same arguments as in the proofof Proposition 4.4, replacingM with A, the quadratic increments with the positive(resp. negative) increments, and the quadratic variation with the positive (resp.negative) variation of an adapted process.

We recall a definition and a result from [2]. We assume for now that we aregiven a fixed stochastic basis fulfilling the usual conditions, and a non-decreasingfunction V .

Notation 4.15. We denote H2,V := M ∈ H20|d⟨M⟩ ≪ dV and H2,⊥V := M ∈

H20|d⟨M⟩ ⊥ dV .

Proposition 3.6 in [2] states the following.

Proposition 4.16. H2,V and H2,⊥V are orthogonal sub-Hilbert spaces of H20 and

H20 = H2,V ⊕⊥ H2,⊥V . Moreover, any element of H2,V

loc is strongly orthogonal to

any element of H2,⊥Vloc .

For any M ∈ H20, we denote by MV its projection on H2,V .

We can now finally establish the main result of the present note.

Proposition 4.17. Let V be a continuous non-decreasing function. Let M,N betwo square integrable MAFs, and assume that the AF ⟨N⟩ is absolutely continuouswith respect to V . There exists a function v ∈ B([0, T ] × E,R) such that for any

(s, x), ⟨Ms,x, Ns,x⟩ =∫ ·∨ss

v(r,Xr)dVr.

Proof. By Corollary 4.11, there exists a bounded variation AF with L1 valuesdenoted ⟨M,N⟩ such that under any Ps,x, the cadlag version of ⟨M,N⟩ is⟨Ms,x, Ns,x⟩. By Proposition 4.14, there exists an increasing AF with L1 val-ues denoted Pos(⟨M,N⟩) (resp. Neg(⟨M,N⟩)) such that under any Ps,x, thecadlag version of Pos(⟨M,N⟩) (resp. Neg(⟨M,N⟩)) is Pos(⟨Ms,x, Ns,x⟩) (resp.Neg(⟨Ms,x, Ns,x⟩)). We fix some (s, x) and the associated probability Ps,x. Since


⟨N⟩ is absolutely continuous with respect to V , comparing Definition 4.1 andNotation 4.15 we have Ns,x ∈ H2,V . Therefore by Proposition 4.16 we have

⟨Ms,x, Ns,x⟩ = ⟨(Ms,x)V , Ns,x⟩= 1

4 ⟨(Ms,x)V +Ns,x⟩ − 1

4 ⟨(Ms,x)V −Ns,x⟩. (4.29)

Since both processes 14 ⟨(M

s,x)V + Ns,x⟩, 14 ⟨(M

s,x)V − Ns,x⟩ are increasing and

starting at zero, we have Pos(⟨Ms,x, Ns,x⟩) = 14 ⟨(M

s,x)V +Ns,x⟩ and

Neg(⟨Ms,x, Ns,x⟩) = 1

4⟨(Ms,x)V −Ns,x⟩.

Now since (Ms,x)V + Ns,x and (Ms,x)V − Ns,x belong to H2,V , we have shownthat dPos(⟨Ms,x, Ns,x⟩) ≪ dV and dNeg(⟨Ms,x, Ns,x⟩) ≪ dV in the sense ofstochastic measures.

Since this holds for all (s, x) Proposition 4.13, insures the existence of twofunctions v+, v− in B([0, T ]×E,R) such that for any (s, x), Pos(⟨Ms,x, Ns,x⟩) =∫ ·∨ss

v+(r,Xr)dVr and Neg(⟨Ms,x, Ns,x⟩) =∫ ·∨ss

v−(r,Xr)dVr. The conclusionnow follows setting v = v+ − v−.

Acknowledgments. The authors are grateful to the Referee for having care-fully checked the paper. The research of the first named author is supported bya PhD fellowship (AMX) of the Ecole Polytechnique. The paper was partiallywritten during a stay of the second named author at Bielefeld University, SFB1283 (Mathematik).

References

1. Aliprantis, C. D. and Border, K. C.: Infinite-dimensional Analysis. Springer-Verlag, Berlin,second edition, 1999. A hitchhiker’s guide.

2. Barrasso, A. and Russo, F.: Backward Stochastic Differential Equations with no driving

martingale, Markov processes and associated Pseudo Partial Differential Equations. (2017).Preprint, hal-01431559, v2.

3. Barrasso, A. and Russo, F.: BSDEs with no driving martingale, Markov processes and associ-ated Pseudo Partial Differential Equations. Part II: Decoupled mild solutions and Examples.

(2017). Preprint, hal-01505974.4. Blumenthal, R. M., Getoor, R. K. and McKean, H. P.Jr: Markov processes with identical

hitting distributions. Bull. Amer. Math. Soc., 68 (1962), 372–373.5. Dellacherie, C. and Meyer, P.-A.: Probabilites et Potentiel, volume A. Hermann, Paris, 1975.

Chapitres I a IV.6. Dellacherie, C. and Meyer, P.-A.: Probabilites et potentiel. Chapitres V a VIII, volume

1385 of Actualites Scientifiques et Industrielles [Current Scientific and Industrial Topics].Hermann, Paris, revised edition, 1980. Theorie des martingales. [Martingale theory].

7. Dellacherie, C. and Meyer, P.-A.: Probabilites et potentiel. Chapitres XII–XVI. Publicationsde l’Institut de Mathematiques de l’Universite de Strasbourg [Publications of the Mathemat-ical Institute of the University of Strasbourg], XIX. Hermann, Paris, second edition, 1987.

Theorie des processus de Markov. [Theory of Markov processes].8. Dynkin, E. B.: Osnovaniya teorii markovskikh protsessov. Teorija Verojatnosteı i

Matematiceskaja Statistika. Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1959.9. Dynkin, E. B.: Additive functionals of Markov processes and stochastic systems. In Annales

de l’Institut Fourier, volume 25 (1975), 177–200.


10. Dynkin, E. B.: Markov Processes and Related Problems of Analysis, volume 54 of London

Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge-NewYork, 1982.

11. Ethier, S. N. and Kurtz, T. G.: Markov processes. Wiley Series in Probability and Mathe-matical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New

York, 1986. Characterization and convergence.12. Flandoli, F., Russo, F., andWolf, J.: Some SDEs with distributional drift. I. General calculus.

Osaka J. Math., 40(2) (2003), 493–542.13. Hsu, E. P.: Stochastic Analysis on Manifolds, volume 38. American Mathematical Soc., 2002.

14. Jacod, J.: Calcul Stochastique et Problemes de Martingales, volume 714 of Lecture Notes inMathematics. Springer, Berlin, 1979.

15. Jacod, J. and Shiryaev, A. N.: Limit Theorems for Stochastic Processes, volume 288 ofGrundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical

Sciences]. Springer-Verlag, Berlin, second edition, 2003.16. Meyer, P.-A.: Fonctionelles multiplicatives et additives de Markov. Ann. Inst. Fourier

(Grenoble), 12 (1962), 125–230.17. Pardoux, E. and Rascanu, A.: Stochastic Differential Equations, Backward SDEs, Partial

Differential Equations, volume 69 of Stochastic Modelling and Applied Probability. Springer,Cham, 2014.

18. Protter, P. E.: Stochastic Integration and Differential Equations, volume 21 of Applica-

tions of Mathematics (New York). Springer-Verlag, Berlin, second edition, 2004. StochasticModelling and Applied Probability.

19. Rozkosz, A.: Weak convergence of diffusions corresponding to divergence form operators.Stochastics Stochastics Rep., 57(1-2) (1996), 129–157.

20. Stroock, D. W.: Diffusion processes associated with Levy generators. Z. Wahrscheinlichkeit-stheorie und Verw. Gebiete, 32(3) (1975), 209–244.

21. Stroock, D. W. and Varadhan, S. R. S.: Multidimensional Diffusion processes. Classics inMathematics. Springer-Verlag, Berlin, 2006. Reprint of the 1997 edition.

Adrien Barrasso: ENSTA ParisTech, UMA, F-91120 Palaiseau and Ecole Polytech-

nique, F-91128 Palaiseau, FranceE-mail address: [email protected]

Francesco Russo: ENSTA ParisTech, UMA, F-91120 Palaiseau, FranceE-mail address: [email protected]

PERPETUAL INTEGRAL FUNCTIONALS OF BROWNIAN

MOTION AND BLOWUP OF SEMILINEAR SYSTEMS OF

SPDES

EUGENIO GUERRERO AND JOSE ALFREDO LOPEZ-MIMBELA*

Abstract. We calculate the probability density of perpetual integral func-tionals of the form

∫∞0 e−((σWs−µs)+x)1(σWs−µs)+x≥0ds, x ≥ 0, where

σ > 0 and µ > 0 are constants and Wt, t ≥ 0 is a one-dimensionalBrownian motion; we achieve this by a direct computation of the potentialmeasure of Brownian motion with drift. By means of the functionals above

we obtain bounds for the blowup times of systems of the form dui(t, x) =

[−(−∆)α/2ui(t, x) + Gi(u3−i(t, x))] dt + κiui(t, x) dWt on a bounded do-main with Dirichlet boundary condition and nonnegative initial values, where

0 < α ≤ 2, κi ≥ 0 is constant and Gi(z) ≥ z1+βi for z ≥ 0 with βi > 0,i = 1, 2.

1. Introduction

Let D ⊂ Rd be a bounded smooth domain, and let κ1, κ2 ∈ R, be given con-stants. Denote by Wt, t ≥ 0 a one-dimensional standard Brownian motiondefined in some probability space (Ω,F ,P), and let f1, f2 ∈ C2(D) be two posi-tive functions. In [6] lower and upper bounds for the explosion time of positivesolutions of the semilinear system of SPDEs

du1(t, x) = [(∆ + V1)u1 (t, x) + up2 (t, x)] dt+ κ1u1 (t, x) dWt,

du2(t, x) = [(∆ + V2)u2 (t, x) + uq1 (t, x)] dt+ κ2u2 (t, x) dWt, (1.1)

ui(0, x) = fi(x) ≥ 0, x ∈ D,

ui(t, x) = 0, t ≥ 0, x ∈ Rd \D, i = 1, 2,

were obtained in the case Vi = λ1 + κ2i /2, i = 1, 2, where λ1 > 0 is the firsteigenvalue of the Laplacian on D and p ≥ q > 1. It was shown that there existrandom times ϱ∗∗, ϱ

∗∗ such that ϱ∗∗ ≤ ϱ ≤ ϱ∗∗, where ϱ is the explosion time of(1.1) and the laws of ϱ∗∗ and ϱ∗∗ are given, respectively, in terms of exponentialfunctionals of the forms∫ t

0

(eaWr ∧ ebWr

)dr and

∫ t

0

(eaWr ∨ ebWr

)dr, t ≥ 0, (1.2)

Received 2017-9-12; Communicated by Hui-Hsiung Kuo.

2010 Mathematics Subject Classification. Primary 35R60, 60H15; Secondary 74H35.Key words and phrases. Blowup, explosion time, perpetual integral functional, potential

measure.

*Research supported in part by CONACyT (Mexico), Grant No. 257867.

335



336 EUGENIO GUERRERO AND JOSE ALFREDO LOPEZ-MIMBELA

for certain real constants a, b. Our aim in this article is to obtain lower and upperbounds for the explosion time of positive solutions of the system of SPDEs

du1(t, x) = [∆αu1 (t, x) +G1(u2(t, x))] dt+ κ1u1 (t, x) dWt,

du2(t, x) = [∆αu2 (t, x) +G2 (u1(t, x))] dt+ κ2u2 (t, x) dWt, (1.3)

ui(0, x) = fi(x) ≥ 0, x ∈ D,

ui(t, x) = 0, t ≥ 0, x ∈ Rd \D, i = 1, 2.

Here, ∆α is the fractional power −(−∆)α/2 of the Laplacian, where 0 < α ≤ 2,and Gi is a locally Lipschitz positive function such that

Gi(z) ≥ z1+βi , z ≥ 0, (1.4)

with βi > 0, i = 1, 2. We assume (1.4) in Section 3.1 only; it is replaced by (3.14)in Section 3.2. We refer to [7] for definitions of blow-up times, and for types ofsolutions of SPDEs. Equations and systems of the above kind arise as mathemati-cal models describing processes of diffusion of heat and burning in two-componentcontinuous media, where the functions u1, u2 are treated as temperatures of in-teracting components in a combustible mixture. Hence, it is natural and relevantto investigate properties of positive solutions of such equations. Since we do notassume Gi to be Lipschitz, i = 1, 2, blowup of the solution of (1.3) in finite timecannot be left out. One of the main contributions of this work is to show that thereare random times τ∗∗ and τ∗∗ such that τ∗∗ ≤ τ ≤ τ∗∗, where τ is the explosiontime of (1.3). In this case, the distributions of the random times τ∗∗ and τ∗∗ aregiven in terms of functionals of the form∫ t

0

(eaWs ∧ ebWs

)e−Msdr and

∫ t

0

(eaWs ∨ ebWs

)e−µsds (1.5)

for some positive constants a, b, M and µ, which depend on the parameters of thesystem (1.3). Notice that the functionals (1.2) are a special case of (1.5), hencethe present paper can be considered as a generalization and an extension of [6].Although the laws of the functionals (1.5) are not given explicitly in this paper,we find random times τ′′ and τ

′′ such that τ′′ ≤ τ∗∗ and τ∗∗ ≤ τ ′′. The randomtimes τ ′′ and τ′′ are given in terms of random functionals of the form

F1(t) =

∫ t

0

e−(σWs−µs)1σWs−µs≥0dr and F2(t) =

∫ t

0

eσWs−µsds, t ≥ 0,

respectively, where σ and µ are certain constants. The function F2 is knownas Dufresne’s functional and the distribution of its perpetual version F2(∞) wascomputed in [8] for µ > 0. The density function of F2(t) for 0 ≤ t < ∞ wasobtained by M. Yor using techniques based on hitting times of Bessel processes;see [17], [3] and [13]. The function F1 is known as one-sided Dufresne’s functional.We believe that the law of its perpetual version could be obtained by the methodof hitting times of Bessel processes as in the case of F2, or else using the methodof Pintoux and Privault [12]. In the present work we calculate the probabilitydensity function of F1(∞) by a straight analytical approach based on the explicitcomputation of the potential measure of the process Xt = σWt − µt, t ≥ 0. This

PERPETUAL FUNCTIONALS AND BLOWUP 337

allows us to obtain a related integral equation for the function

H (x, z) = E[exp

(−z∫ ∞

0

e−(Xs+x)1Xs+x≥0ds

)], x ≥ 0, z > 0,

(which gives as a special case the Laplace transform of F1(∞)), and upon solvingit we obtain an explicit expression for H. By inverting the transform H we getthe distribution of the perpetual functional F1(∞) which is needed further toobtain a lower bound for the probability of explosion in finite time. This is thesubject of Section 2. With the aim of getting suitable sub- and supersolutionsof (1.3) –from which we will obtain upper and lower bounds for τ–, in Section3 we transform system (1.3) into a related system of random partial differentialequations. This procedure is similar to the one performed in [6] and is inspired ina classical result of Doss [5]. In Section 3 we also obtain upper and lower boundsfor the explosion time τ . In Section 4 we give explicit non-trivial bounds for theprobability of explosion in finite time of positive solutions of system (1.3), underthe assumptions that β1 = β2 and the initial values are of the form fi(x) = Liψ(x),x ∈ D, with Li > 0, i = 1, 2, where ψ is the eigenfunction corresponding to thefirst eigenvalue of ∆α on D. Such bounds depend on the functionals we found inSection 3.

2. An Exponential Functional of Brownian Motion

Let Wt, t ≥ 0 be a one-dimensional standard Brownian motion. Let σ andµ be positive constants. It is well known (see e.g. [8]) that Dufresne’s functional∫∞0eσWs−µsds has the following distribution for all c ≥ 0 :

P(∫ ∞

0

eσWs−µsds > c

)=γ(2µσ2 ,

2σ2c

)Γ(2µσ2

) , (2.1)

where γ (a, x) =∫ x0e−ssa−1ds and Γ (a) = γ (a,∞) for all a > 0 and x ≥ 0.

Let Xt = σWt − µt, t ≥ 0. The motivation of this section is to study, from ananalytical point of view, some distributional properties of the exponential func-tional ∫ ∞

0

e−(Xt+x)1Xt+x≥0dt, x ≥ 0.

This kind of functionals, also named one-sided variants of Dufresne’s functional,emerges for instance in the problem of explosion in finite time of systems of SPDEs.In particular we calculate explicitly its Laplace transform and its distribution atx = 0. Recall (see [2]) that the potential measure of the process Xt, t ≥ 0 isthe Borel measure U defined by

U (B) =

∫ ∞

0

P (Xt ∈ B) dt, B ∈ B (R) ,

where B (R) stands for the Borel σ-algebra in R = (−∞,∞).

Lemma 2.1. The measure U is absolutely continuous with respect to the Lebesguemeasure, and the density function of U is given by

u (x) =1

µ1(−∞,0) (x) +

1

µe−

2µ

σ2x1[0,∞) (x) , x ∈ R. (2.2)


Proof. First note that the transition probability of Xt, t ≥ 0 is given by

p (t, x) =1√

2πσ2texp

[− (x+ µt)

2

2σ2t

], x ∈ R, t > 0.

From [14, page 242] we know that

u (x) =

∫ ∞

0

p (t, x) dt

=

√2

πσ2µ

∫ ∞

0

exp

[−(√

µ

2σ2

x

s+

√µ

2σ2s

)2]ds, x ∈ R,

where we have used the change of variables s =√µt to obtain the second equality.

Now we note that for all s > 0,√2

πσ2µe−(

√µ

2σ2xs+

√µ

2σ2s)

2

=e−

µ

σ2(|x|+x)

2µ

[− 2√

πe−(

√µ

2σ2|x|s −

√µ

2σ2s)

2(−√

µ

2σ2

|x|s2

−√

µ

2σ2

)+ e

2µ

σ2|x| 2√

πe−(

√µ

2σ2|x|s +

√µ

2σ2s)

2(−√

µ

2σ2

|x|s2

+

√µ

2σ2

)]. (2.3)

Integrating both sides of (2.3) with respect to s, we get for all x ∈ R,

u (x) =e−

µ

σ2(|x|+x)

2µ

[− 2√

π

∫ ∞

0

e−(√

µ

2σ2|x|s −

√µ

2σ2s)

2(−√

µ

2σ2

|x|s2

−√

µ

2σ2

)ds

+ e2µ

σ2|x| 2√

π

∫ ∞

0

e−(√

µ

2σ2|x|s +

√µ

2σ2s)

2(−√

µ

2σ2

|x|s2

+

√µ

2σ2

)ds

].

Performing the change of variables

t =

√µ

2σ2

|x|s

−√

µ

2σ2s and t =

√µ

2σ2

|x|s

+

√µ

2σ2s

in the integrals of the right hand side renders

u (x) =e−

µ

σ2(|x|+x)

2µ

[−erf

(√µ

2σ2

|x|s

−√

µ

2σ2s

)+ e

2µ

σ2|x|erf

(√µ

2σ2

|x|s

+

√µ

2σ2s

)]∣∣∣∣∞0

,

where

erf (z) =2√π

∫ z

0

e−s2

ds, z ∈ R,

is the error function. Since erf (∞) = 1 and erf (−∞) = −1, it follows that

u (x) =e−

2µ

σ2x

2µ

[−erf (−∞) + e

2µ

σ2xerf (∞) + erf (∞)− e

2µ

σ2xerf (∞)

]=

1

µe−

2µ

σ2x,

for all x ≥ 0. Similarly, if x < 0 we conclude that u (x) = 1/µ and the resultfollows.


Define

H (x, z) = E[exp

(−z∫ ∞

0

e−(Xs+x)1Xs+x≥0 ds

)]for all x ≥ 0, z ∈ C.

Lemma 2.2. For all x ≥ 0 and z ∈ C, H (x, z) satisfies the integral equation

H (x, z) = 1− µ−1ze2µ

σ2x

∫ ∞

x

e−(1+2µ

σ2)uH (u, z) du− µ−1z

∫ x

0

e−uH (u, z) du.

(2.4)

Proof. For simplicity of notation, for any fixed z ∈ C, let fz(x) = −ze−x1x≥0,x ∈ R. Define the function

vt (x, z) = E[exp

(∫ t

0

fz(Xs + x)ds

)], t ≥ 0, x ≥ 0.

Using that∫ t

0

exp

(∫ t

s

fz(Xu + x)du

)fz(Xs + x)ds

= −∫ t

0

d

ds

[exp

(∫ t

s

fz(Xu + x)du

)]ds = exp

(∫ t

0

fz(Xu + x)du

)− 1,

from the Dominated Convergence Theorem we get

vt(x, z) = 1 + E[∫ t

0

exp

(∫ t

s

fz(Xu + x)du

)fz(Xs + x)ds

]= 1 +

∫ t

0

E[exp

(∫ t

s

fz(Xu + x)du

)fz(Xs + x)

]ds. (2.5)

Since fz(Xs + x) is measurable with respect to σ(Xr, 0 ≤ r ≤ s), 0 ≤ s ≤ t, then

E[exp

(∫ t

s

fz(Xu + x)du

)fz(Xs + x)

]= E

[fz(Xs + x)E

[exp

(∫ t

s

fz(Xu + x)du

)∣∣∣∣σ(Xr, 0 ≤ r ≤ s)

]]. (2.6)

Due to the independence of increments property of Xt, t ≥ 0 we get

E[exp

(∫ t

s

fz(Xu + x)du

)∣∣∣∣σ(Xr, 0 ≤ r ≤ s)

]= E

[exp

(∫ t

s

fz(Xu −Xs +Xs + x)du

)∣∣∣∣σ(Xr, 0 ≤ r ≤ s)

]= h(Xs + x), (2.7)

where the function h is defined by

h(y) = E[exp

(∫ t

s

fz(Xu −Xs + y)du

)].


Due to stationarity of increments of Xt, t ≥ 0, we obtain that

h(y) = E[exp

(∫ t−s

0

fz(Xu + y)du

)]= vt−s(y, z). (2.8)

Plugging (2.6), (2.7) and (2.8) into (2.5) we finally get

vt(x, z) = 1 +

∫ t

0

E [fz(Xs + x)vt−s(Xs + x, z)] ds

= 1 + E[∫ t

0

fz(Xs + x)vt−s(Xs + x, z)ds

]= 1− z

∫Re−(x+y)1[0,∞) (x+ y)

(∫ t

0

vt−s (x+ y, z)P (Xs ∈ dy) ds

).

Since the improper integral∫∞0e−(Xs+x)1Xs+x≥0 ds is a.s. finite due to [9,

Theorem 1.4], using dominated convergence we get

vt (x, z) → H (x, z) as t→ ∞.

The fact that

0 ≤∣∣vt−s (x+ y, z)1[0,t] (s)

∣∣ ≤ 1

for all s ≥ 0 implies, for all x ≥ 0,∣∣∣∣∫Re−(x+y)1[0,∞) (x+ y)

(∫ ∞

0

vt−s (x+ y, z)1[0,t] (s)P (Xs ∈ dy) ds

)∣∣∣∣≤∫Re−(x+y)1[0,∞) (x+ y)U (dy) =

1− e−x

µ+

σ2e−x

µσ2 + 2µ2≤ 1

µ+

σ2

µσ2 + 2µ2.

Using again dominated convergence we get that for every z ∈ C and every x ≥ 0the function H(·, z) satisfies the integral equation

H (x, z) = 1− z

∫Re−(x+y)1[0,∞) (x+ y)H (x+ y, z)U (dy) . (2.9)

From (2.2) and (2.9) it follows that for every z ∈ C and every x ≥ 0

H (x, z) = 1− z

∫Re−(x+y)1[0,∞) (x+ y)H (x+ y, z)µ−11(−∞,0) (y) dy

− z

∫Re−(x+y)1[0,∞) (x+ y)H (x+ y, z)µ−1e−

2µ

σ2y1[0,∞) (y) dy

= 1− µ−1z

∫ 0

−xe−(x+y)H (x+ y, z) dy

− µ−1z

∫ ∞

0

e−(x+y)H (x+ y, z) e−2µ

σ2ydy

= 1− µ−1ze2µ

σ2x

∫ ∞

x

e−(1+2µ

σ2)uH (u, z) du− µ−1z

∫ x

0

e−uH (u, z) du.


Theorem 2.3. Let θ ∈ C be such that |θ| < 1, and let

I (x, u) = e2µ

σ2xe−(1+

2µ

σ2)u1[x,∞) (u) + e−u1[0,x) (u) , x, u ≥ 0.

Then the integral equation

g (x) = 1− θ

∫ ∞

0

I (x, u) g (u) du (2.10)

possesses a unique solution

g (x) =∑n≥0

(−θ)n ψn (x) ∈ Cb(R+),

where

ψ0 (x) = 1, ψn+1 (x) =

∫ ∞

0

I (x, u)ψn (u) du, n ≥ 0, x ≥ 0.

Proof. Consider the Banach space (Cb (R+) , ∥·∥∞). We have that∫ ∞

0

I (x, u) du = 1− 2µ

σ2 + 2µe−x,

which implies that the function

h (x) := 1− θ

∫ ∞

0

I (x, u) g (u) du, x ∈ R+,

satisfies h ∈ Cb (R+) for all g ∈ Cb (R+). Now we prove that the operator T :Cb (R+) → Cb (R+), defined by T (g) = h, is a contraction mapping. In fact, forg1, g2 ∈ Cb (R+),

∥T (g1)− T (g2)∥∞ = |θ|∥∥∥∥∫ ∞

0

I (·, u) g1 (u) du−∫ ∞

0

I (·, u) g2 (u) du∥∥∥∥∞

≤ |θ|∥∥∥∥∫ ∞

0

I (·, u) du∥∥∥∥∞

∥g1 − g2∥∞ = |θ| ∥g1 − g2∥∞ ,

i.e., T is a contraction mapping. From the Banach fixed point theorem it followsthat (2.10) has a unique solution. To prove the power series representation ofg first we note that ∥ψn∥∞ ≤ 1 for all n ≥ 0, which can be easily proved byinduction. Then under the assumption |θ| ∈ [0, 1), the series∑

n≥0

(−θ)n ψn

is absolutely and uniformly convergent. By Fubini’s theorem we finally get that

1− θ

∫ ∞

0

I (x, u)∑n≥0

(−θ)n ψn (u) du = 1 +∑n≥0

(−θ)n+1∫ ∞

0

I (x, u)ψn (u) du

= 1 +∑n≥0

(−θ)n+1ψn+1 (x) =

∑n≥0

(−θ)n ψn (x) .

Thereforeg (x) =

∑n≥0

(−θ)n ψn (x)

for all x ≥ 0.


From Lemma 2.2 and Theorem 2.3 we deduce one of the main results of thissection.

Theorem 2.4. For all x ≥ 0 and all z ∈ C such that |z|µ−1 < 1, the functionH (x, z) is the unique solution of the integral equation

F (x, z) = 1− µ−1ze2µ

σ2x

∫ ∞

x

e−(1+2µ

σ2)uF (u, z) du− µ−1z

∫ x

0

e−uF (u, z) du.

(2.11)

In order to get a closed expression for H, we proceed by induction over n ≥ 0to prove that

ψn+1 (x) =

n∑k=1

Bkψn+1−k (x) +Bn+1

(1−

2µσ2

n+ 1 + 2µσ2

e−(n+1)x

), (2.12)

where

Bk :=(−1)

k−1Γ(2µσ2

) (2µσ2

)kk!Γ

(k + 2µ

σ2

) , k ∈ N.

For n = 0, under the convention∑0k=1 ≡ 0 and the fact that B1 = 1, we get

ψ1 (x) =

∫ ∞

0

(e

2µ

σ2xe−(1+

2µ

σ2)u1[x,∞) (u) + e−u1[0,x) (u)

)du = 1−

2µσ2

1 + 2µσ2

e−x,

which shows that (2.12) holds for n = 0. Assume that (2.12) is true for somen ≥ 0. Then

ψn+1 (x) =

∫ ∞

0

I (x, u)ψn (u) du

=

∫ ∞

0

I (x, u)

(n−1∑k=1

Bkψn−k (u) +Bn

(1−

2µσ2

n+ 2µσ2

e−nu

))du

=n−1∑k=1

Bk

∫ ∞

0

I (x, u)ψn−k (u) du

+Bn

(∫ ∞

0

I (x, u) du−2µσ2

n+ 2µσ2

∫ ∞

0

I (x, u) e−nudu

)

=n−1∑k=1

Bkψn+1−k (x) +Bnψ1 (x)

−2µσ2

n+ 2µσ2

Bn

(1

n+ 1−

2µσ2

(n+ 1)(n+ 1 + 2µ

σ2

)e−(n+1)x

)

=

n∑k=1

Bkψn+1−k (x) +Bn+1

(1−

2µσ2

n+ 1 + 2µσ2

e−(n+1)x

),


where in the second equality we have used the induction hypothesis, the definitionof ψn for the fourth one and the fact that

Bn+1 = −2µσ2

(n+ 1)(n+ 2µ

σ2

)Bnfor the last equality. This proves (2.12). Moreover, notice that∑

n≥1

(−µ−1z

)nψn (x)

=∑n≥1

(−µ−1z

)n(n−1∑k=1

Bkψn−k (x)

+ Bn

(1−

2µσ2

n+ 2µσ2

e−nx

))=

∑k≥1

Bk∑

n≥k+1

(−µ−1z

)nψn−k (x)

+∑n≥1

(−µ−1z

)nBn

(1−

2µσ2

n+ 2µσ2

e−nx

)=

∑k≥1

Bk∑j≥1

(−µ−1z

)j+kψj (x)

+∑n≥1

(−µ−1z

)nBn

(1−

2µσ2

n+ 2µσ2

e−nx

)=

∑k≥1

Bk(−µ−1z

)k∑j≥1

(−µ−1z

)jψj (x)

+∑n≥1

(−µ−1z

)nBn

(1−

2µσ2

n+ 2µσ2

e−nx

),

from which we conclude that∑n≥1

(−µ−1z

)nψn (x)

=

∑n≥1

(−µ−1z

)nBn

(1−

2µ

σ2

n+ 2µ

σ2

e−nx)

1−∑n≥1 (−µ−1z)

nBn

,

and therefore we get

H (x, z) =

1−∑n≥1

(−µ−1z

)nBn

(2µ

σ2

n+ 2µ

σ2

)e−nx

1−∑n≥1 (−µ−1z)

nBn

. (2.13)


From the definition of Bn,∑n≥1

(−µ−1z

)nBn

(2µσ2

n+ 2µσ2

)e−nx (2.14)

=∑n≥1

(−µ−1ze−x

)n (−1)n−1

Γ(2µσ2

) (2µσ2

)nn!Γ

(n+ 2µ

σ2

) (2µσ2

n+ 2µσ2

)

= 1− 2µ

σ2Γ

(2µ

σ2

)∑n≥0

(2zσ2 e

−x)nn!Γ

(n+ 1 + 2µ

σ2

)

= 1− 2µ

σ2

(2z

σ2e−x

)− µ

σ2

Γ

(2µ

σ2

)∑n≥0

(2( 2zσ2e−x)

1/2

2

)2n+ 2µ

σ2

n!Γ(n+ 1 + 2µ

σ2

)= 1− 2µ

σ2

(2z

σ2e−x

)− µ

σ2

Γ

(2µ

σ2

)I 2µ

σ2

(2

(2z

σ2e−x

)1/2), (2.15)

where

Iν (z) :=∑k≥0

(z2

)2k+νk!Γ (k + 1 + ν)

, z ∈ C,

is the modified Bessel function of the first kind of order ν ∈ R. Similarly, it canbe shown that∑

n≥1

(−µ−1z

)nBn = 1− Γ

(2µ

σ2

)(2z

σ2

) 12−

µ

σ2

I 2µ

σ2−1

(2

(2z

σ2

)1/2). (2.16)

Plugging (2.15) and (2.16) into (2.13) we get

H (x, z) = 2µσ−1 (2z)−1/2

eµ

σ2xI 2µ

σ2

(2σ−1 (2z)

1/2e−

x2

)I 2µ

σ2−1

(2σ−1 (2z)

1/2) , (2.17)

for all x ≥ 0 and z ∈ C such that |z|µ−1 < 1. In particular we obtain

Theorem 2.5. The equality

E[exp

(−z∫ ∞

0

e−Xt1Xt≥0dt

)]=

4µI 2µ

σ2

(√8zσ

)σ√8zI 2µ

σ2−1

(√8zσ

) (2.18)

holds for every z ∈ C such that |z|µ−1 < 1.

Let F be the distribution function of the random variable∫ ∞

0

e−Xt1Xt≥0 dt.


Letj 2µ

σ2−1,n

n≥1

be the increasing sequence of all positive zeros of the Bessel

function of the first kind of order 2µσ2 − 1 > −1, and let

J 2µ

σ2−1 (z) :=

∑m≥0

(−1)m ( z

2

)2m+ 2µ

σ2−1

m!Γ(m+ 2µ

σ2

) , z ∈ C.

From the fact that

J 2µ

σ2(z)

J 2µ

σ2−1 (z)

= −2z∑n≥1

(z2 − j22µ

σ2−1,n

)−1

, z ∈ C \±j 2µ

σ2−1,n

n≥1

,

(see [10, formula 7.9(3)]) and the relation Jν (zi) = iνIν (z), which holds for allν, z ∈ R, it follows that

z−1/2I 2µ

σ2

(z1/2

)I 2µ

σ2−1

(z1/2

) = 2∑n≥1

1

z + j22µσ2

−1,n

, z ∈ C \−j22µ

σ2−1,n

n≥1

. (2.19)

Notice that the function

z 7→ z−1/2 I2µ/σ2(z1/2)

I2µ/σ2−1(z1/2), z ∈ C,

has no poles in the region

w ∈ C : Re w > 0, |w| < µ .Using an analytic continuation argument we conclude that

E[exp

(−z∫ ∞

0

e−Xt1Xt≥0dt

)]=

4µI 2µ

σ2

(√8zσ

)σ√8zI 2µ

σ2−1

(√8zσ

) ,for all z ∈ w ∈ C : Re w > 0. In particular we get that the Laplace transformof the random variable ∫ ∞

0

e−Xt1Xt≥0dt

is given, for all z ≥ 0, by

E[exp

(−z∫ ∞

0

e−Xt1Xt≥0dt

)]=

8µ

σ2

∑n≥1

18zσ2 + j22µ

σ2−1,n

=8µ

σ2

∑n≥1

∫ ∞

0

e−zy

σ2

8e−(σ2

8 j22µ

σ2−1,n

)y

dy=

∫ ∞

0

e−zy

µ∑n≥1

e−(σ2

8 j22µ

σ2−1,n

)y

dy,where we used the fact that ∑

n≥1

1

j2ν,n=

1

4(ν + 1)


for any ν > −1 (see [4, formula (32)]). In this way we have proved the followingresult.

Theorem 2.6. F is absolutely continuous with respect to the Lebesgue measure.Furthermore, if y ≥ 0 then

F (dy) = µ

∑n≥1

exp

−(σ2

8j22µσ2

−1,n

)y

dy. (2.20)

3. Bounds for the Explosion Time

In this section we obtain upper and lower bounds for the explosion time of thesemilinear system (1.3). For this, we first construct a suitable subsolution of (1.3)by means of the change of variables

vi (t, x) := exp −κiWtui (t, x) , t ≥ 0, x ∈ D, i = 1, 2,

which transforms a weak solution (u1, u2) of (1.3) into a weak solution of a systemof random parabolic PDEs. Proceeding as in [6] (see also [5]) one can see that thefunction (v1 (t, x) , v2 (t, x)) is a weak solution of the system of RPDEs

∂

∂tv1 (t, x) =

(∆αv1 (t, x)−

κ212v1 (t, x)

)+ e−κ1WtG1

(eκ2Wtv2 (t, x)

),

∂

∂tv2 (t, x) =

(∆αv2 (t, x)−

κ222v2 (t, x)

)+ e−κ2WtG2

(eκ1Wtv1 (t, x)

), (3.1)

vi (0, x) = fi (x) ≥ 0, x ∈ D,

vi (t, x) = 0, t ≥ 0, x ∈ Rd \D, i = 1, 2,

with the same assumptions as in (1.3). Notice that vi(t, ·) is non-negative on Dfor each t ≥ 0 and i = 1, 2, which follows from the Feynman-Kac representationof (3.1); see e.g. [1]. Hence

ui(t, ·) = expκiWtvi(t, ·)is also non-negative on D for each t ≥ 0 and i = 1, 2. Moreover, it is clear thatif τ is the blowup time of system (1.3), then τ is also the blowup time of system(3.1). Let λ and ψ be, respectively, the first eigenvalue and eigenfunction of ∆α

in D, with ψ normalized so that∫Dψ (x) dx = 1.

3.1. An upper bound for the explosion time. In order to get an upper boundfor the explosion time τ , we first show that the function

t 7→∫D

v (t, x)ψ (x) dx, t > 0,

satisfies the differential inequality

d

dt

∫D

vi (t, x)ψ (x) dx ≥ −(λ+

κ2i2

)∫D

vi (t, x)ψ (x) dx

+ e((1+βi)κ3−i−κi)Wt

(∫D

v3−i (t, x)ψ (x) dx

)1+βi

,

(3.2)


for i = 1, 2 and t > 0. In fact, since vi (t, x) is a weak solution of (3.1) and

Gi(z) ≥ z1+βi , z ≥ 0,

then in particular we have∫D

vi (t, x)ψ (x) dx ≥∫D

fi (x)ψ (x) dx+

∫ t

0

∫D

vi (s, x)∆αψ (x) dx ds

− κ2i2

∫ t

0

∫D

vi (s, x)ψ (x) dx ds (3.3)

+

∫ t

0

∫D

e((1+βi)κ3−i−κi)Wsv1+βi3−i (s, x)ψ (x) dx ds.

Since vi and ψ are non-negative in D, by Holder’s inequality we get that∫D

v3−i (s, x)ψ (x) dx =

∫D

v3−i (s, x)ψ1

1+βi (x)ψβi

1+βi (x) dx

≤(∫

D

v1+βi3−i (s, x)ψ (x) dx

) 11+βi

. (3.4)

Using the fact that

∆αψ (x) = −λψ (x) on D,

we finally obtain∫D

vi (t, x)ψ (x) dx ≥∫D

fi (x)ψ (x) dx−(λ+

κ2i2

)∫ t

0

∫D

vi (s, x)ψ (x) dxds

+

∫ t

0

e((1+βi)κ3−i−κi)Ws

(∫D

v3−i (s, x)ψ (x) dx

)1+βi

ds,

whose differential form is (3.2). Using now (3.2) and a comparison theorem (seee.g. [16, Lemma 1.2]), we deduce that the function hi determined by the equation

d

dthi (t) = −

(λ+

κ2i2

)hi (t) + e((1+βi)κ3−i−κi)Wth1+βi3−i (t) ,

hi (0) =

∫D

fi (x)ψ (x) dx,

is a subsolution of vi, i = 1, 2. We define

m = λ+ maxi=1,2

κ2i2

, Mt = min

i=1,2

e((1+βi)κ3−i−κi)Wt

, t ≥ 0,

and consider the system of random ODEs

d

dtzi (t) = −mzi (t) +Mtz

1+βi3−i (t) , zi (0) = hi (0) , i = 1, 2.

By the transformation

yi (t) := emtzi (t) , t ≥ 0, i = 1, 2,

it follows that

d

dtyi (t) = e−mβitMty

1+βi3−i (t) , yi (0) = hi (0) , i = 1, 2. (3.5)


By a comparison argument it follows that

hi (t) ≥ zi (t) , t ≥ 0, i = 1, 2.

For t ≥ 0 we define

E (t) = y1 (t) + y2 (t) with E (0) =

2∑i=1

∫D

fi (x)ψ (x) dx.

We present the main result of this section, where

A := mini=1,2

(1 + βi)κ3−i − κi .

Theorem 3.1. Assume that A > 0 and let τ be the blow-up time of system (1.3).

(1) If β1 = β2, then τ ≤ τ ′, where

τ ′ = inf

t ≥ 0 :

∫ t

0

e−(AWs−mβ1s)1AWs−mβ1s≥0ds ≥ 2β1β−11 (E (0))

−β1

.

(3.6)(2) Suppose β1 > β2 > 0. Let

ϵ0 = min

1,

(h2 (0)

A1/(1+β2)0

)β1−β2

,

(2−(1+β2) (E (0))

1+β2

A0

) β1−β21+β2

,

with

A0 =

(1 + β11 + β2

)− 1+β2β1−β2 β1 − β2

1 + β1.

Assume that

2−β2ϵ0 (E (0))1+β2 − ϵ

1+β1β1−β20 A0 > 0, (3.7)

and let

C0 = 2−β2ϵ0 −ϵ

1+β1β1−β20 A0

(E (0))1+β2

.

Then τ ≤ τ ′′, where

τ ′′ = inf

t ≥ 0 :

∫ t

0

e−(AWs−mβ2s)1AWs−mβ2s≥0ds ≥ C−10 β−1

2 (E (0))−β2

.

(3.8)

Proof. Recall that

x1+β1 + y1+β1 ≥ 2−β1 (x+ y)1+β1

for all x, y ∈ [0,∞). Therefore, from (3.5) we get

d

dtE (t) ≥ 2−β1e−mβ1tMtE

1+β1 (t) .

Using a comparison argument as before, it is clear that I is a subsolution of E,where

d

dtI (t) = 2−β1e−mβ1tMtI

1+β1 (t) , I (0) = E (0) .


The solution of this equation is given by

I (t) =

(I−β1 (0)− 2−β1β1

∫ t

0

e−mβ1sMsds

)− 1β1

, t ∈ [0, τ∗) ,

with

τ∗ := inf

t ≥ 0 :

∫ t

0

e−mβ1sMsds ≥ 2β1β−11 I−β1 (0)

. (3.9)

The inequality τ ≤ τ∗ is clear since I is a subsolution of v1+ v2. There remains toshow the inequality τ∗ ≤ τ ′, where τ ′ is defined in (3.6). This follows easily fromthe fact that

e−mβ1sMs ≥ e−mβ1seAWs1Ws≥0

andAWs −mβ1 ≥ 0 ⊆ Ws ≥ 0

for all s ≥ 0. We conclude that∫ t

0

e−mβ1sMsds ≥∫ t

0

e−(AWs−mβ1s)1AWs−mβ1s≥0ds,

and the assertion follows. Therefore τ ≤ τ ′.We now prove part (2) of the theorem. According to Young’s inequality,

xy ≤ δ−pxp

p+δqyq

q(3.10)

for all x, y ∈ [0,∞), δ > 0 and p, q ∈ (1,∞) such that 1p +

1q = 1. Taking A0 as in

the statement and setting

x = ϵ, y = y1+β2

2 (t) , δ =

(1 + β11 + β2

) 1+β21+β1

and q =1 + β11 + β2

in (3.10), it follows that for all ϵ > 0,

y1+β1

2 (t) ≥ ϵy1+β2

2 (t)− ϵ1+β1β1−β2A0, t ≥ 0.

Using (3.5) we get

d

dtE (t) ≥ e−mβ1tMt

(y1+β2

1 (t) + ϵy1+β2

2 (t)− ϵ1+β1β1−β2A0

). (3.11)

Suppose ϵ ∈ (0, 1]. Using Jensen’s inequality we conclude that

y1+β2

1 (t) + ϵy1+β2

2 (t) ≥ 2−β2

[y1 (t) + ϵ

11+β2 y2 (t)

]1+β2

≥ 2−β2ϵ [y1 (t) + y2 (t)]1+β2 = 2−β2ϵE1+β2 (t) ,

henced

dtE (t) ≥ e−mβ1tMt

(2−β2ϵE1+β2 (t)− ϵ

1+β1β1−β2A0

).

Take ϵ0 as in the statement. We claim that

E (t) ≥ E (0) > 0 for all t ≥ 0.

In fact, let J be the solution of the differential equation

J ′ (t) = e−mβ1tMtf (J (t)) , J (0) = E (0) ,


where

f (x) := 2−β2ϵ0x1+β2 − ϵ

1+β1β1−β20 A0, x ≥ 0.

By comparison E(t) ≥ J(t) for all t ≥ 0, and therefore it suffices to show that

J(t) ≥ E(0), t ≥ 0.

Notice that f is increasing and has only one zero at

x0 = (2β2ϵ1+β2β1−β20 A0)

11+β2 > 0,

with x0 < E(0) due to (3.7). Let

T = inf t > 0 : J(t) < E(0) .

Then T > 0 because J is strictly increasing around 0, and J(t) ≥ E(0) for allt ∈ (0, T ). Suppose that T <∞. Being J continuous on [0, T ] and differentiable on(0, T ), Rolle’s theorem yields that J ′(c) = 0 for some c ∈ (0, T ). Hence J(c) = x0which implies that x0 ≥ E(0). This contradiction says that T = ∞ and

E(t) ≥ E(0) for all t ≥ 0,

which proves the claim. Therefore,

d

dtE (t) ≥ e−mβ1tMtE

1+β2 (t)

2−β2ϵ0 −ϵ

1+β1β1−β20 A0

(E (0))1+β2

.Let C0 be as in the statement and let I be the solution of the equation

d

dtI (t) = e−mβ1tMtI

1+β2 (t)C0, t ∈ [0, τ∗∗) ; I (0) = E (0) ,

where τ∗∗ will be defined below. Then I(t) ≤ E(t). The expression for I is givenin this case by

I (t) =

(I−β2 (0)− C0β2

∫ t

0

e−mβ1sMsds

)− 1β2

,

for all t ∈ [0, τ∗∗), with τ∗∗ given by

τ∗∗ = inf

t ≥ 0 :

∫ t

0

e−mβ1sMsds ≥ C−10 β−1

2 I−β2 (0)

. (3.12)

Taking τ ′′ as in (3.8) and proceeding as in the proof of Part 1, we get τ ≤ τ∗∗ ≤τ ′′.


Remark 3.2. When κ1 = κ2 = 0 and β1 = β2 > 0, from the inequality τ ≤ τ∗ itfollows that

P (τ <∞) ≥ P(1

λ> 2β1I−β1 (0)

)≥ P

(1

λ>

(mini=1,2

∫D

fi (x)ψ (x) dx

)−β1)

=

1 if λ1β1 < min

i=1,2

∫D

fi (x)ψ (x) dx

,

0 otherwise,

which is the deterministic result given in [11].

3.2. A lower bound for the explosion time. Suppose that Yt, t ≥ 0 is aspherically symmetric α-stable process with infinitesimal generator ∆α. Let

τD := inf t > 0 : Yt /∈ Dand consider the killed process

Y Dt , t ≥ 0

given by

Y Dt =

Yt if t < τD

∂ if t ≥ τD,

where ∂ is a cemetery point. Let T ≥ 0 be a random time. Recall that a pair ofFt-adapted random fields

(v1 (t, x) , v2 (t, x)) , x ∈ D, t ≥ 0, i = 1, 2,

is a mild solution of (3.1) in the interval [0, T ] if

vi (t, x) = e−κ2i2 tPDt fi (x) +

∫ t

0

e−κiWre−κ2i2 (t−r)PDt−r

[Gi(eκ3−iWrv3−i (r, x)

)]dr,

(3.13)P-a.s. for all t ∈ (0, T ], i = 1, 2, where

PDt , t ≥ 0

is the semigroup of the process

Y Dt , t ≥ 0. In what follows we will assume that Gi is a locally Lipschitz positive

function such thatGi(z) ≤ z1+βi , z ≥ 0, i = 1, 2. (3.14)

Moreover, we set

A = mini=1,2

(1 + βi)κ3−i − κi and B = maxi=1,2

(1 + βi)κ3−i − κi .

Theorem 3.3. Let β = maxi=1,2

βi and

ϕ(t) = e−(κ1∧κ2)2t/2 max

i=1,2

sups∈[0,t]

∥∥PDs fi∥∥∞, t ≥ 0.

Assume that A > 0. Thenvi(t, x) ≤ ϕ(t)B(t)

for all 0 ≤ t < τ∗, x ∈ D and i = 1, 2, where

B (t) =

(1− β

∫ t

0

(eAWr ∨ eBWr

)maxi=1,2

ϕβi (r)

dr

)− 1β

.


and

τ∗ = inf

t ≥ 0 :

∫ t

0

(eAWr ∨ eBWr

)maxi=1,2

ϕβi (r)

dr ≥ 1

β

. (3.15)

Proof. Notice that B (0) = 1 and

d

dtB (t) =

(eAWt ∨ eBWt

)maxi=1,2

ϕβi (t)

B1+β (t) , t > 0,

hence

B (t) = 1 +

∫ t

0

(eAWr ∨ eBWr

)maxi=1,2

ϕβi (r)

B1+β (r) dr.

Now let V : [0,∞)×D → R be a non-negative continuous function such that

V (t, ·) ∈ C0 (D) , t ≥ 0,

and satisfying

V (t, x) ≤ ϕ(t)B (t) , t ∈ [0, τ∗) , x ∈ D. (3.16)

Define the operator Fi by

Fi (V (t, x)) := e−κ2i2 tPDt fi (x)+

∫ t

0


[Gi(eκ3−iWrV (r, x)

)]dr,

for i = 1, 2. Using (3.14) and that the semigroupPDt , t ≥ 0

preserves positivity

we get

Fi (V (t, x)) ≤ ϕ(t) +

∫ t

0

e((1+βi)κ3−i−κi)Wre−(κ1∧κ2)2

2 (t−r)PDt−r[V 1+βi (r, x)

]dr

≤ ϕ(t) +

∫ t

0

e((1+βi)κ3−i−κi)Wre−(κ1∧κ2)2

2 (t−r)ϕ1+βi (r)B1+βi (r) dr,

where we have used (3.16) to obtain the last inequality. Notice that if t ∈ [0, τ∗)and r ∈ [0, t] then

e−(κ1∧κ2)2

2 (t−r)ϕ (r) ≤ ϕ(t),

and since B(t) ≥ 1,

B1+βi (r) ≤ B1+β(r), 0 ≤ r ≤ t.

Therefore, for all t ∈ [0, τ∗) and x ∈ D,

Fi (V (t, x)) ≤ ϕ(t)

[1 +

∫ t

0

(eAWr ∨ eBWr

)maxi=1,2

ϕβi (r)

B1+β (r) dr

]= ϕ(t)B(t).

Now we will define increasing sequences which will converge to the mild solutionof (3.1). Let

v1,0 (t, x) = e−κ212 tPDt f1 (x) , v2,0 (t, x) = e−

κ222 tPDt f2 (x) , (t, x) ∈ [0, τ∗)×D,

and for any n ≥ 0 define

v1,n+1 (t, x) = F1 (v2,n (t, x)) , v2,n+1 (t, x) = F2 (v1,n (t, x)) ,


for (t, x) ∈ [0, τ∗) × D. To prove that (v1,n (t, x))n≥0 and (v2,n (t, x))n≥0 are

increasing for all t ∈ [0, τ∗) and x ∈ D, note that

vi, (t, x) ≤ e−κ2i2 tPDt fi (x) +

∫ t

0


[Gi(eκ3−iWrv3−i,0 (r, x)

)]dr

= v1i (t, x) , i = 1, 2.

Suppose that, for some n ≥ 0,

vi,n ≥ vi,n−1, i = 1, 2.

Then

vi,n+1(t, x) = Fi (v3−i,n(t, x)) ≥ Fi (v3−i,n−1(t, x)) = vi,n(t, x)

for all (t, x) ∈ [0, τ∗)×D, where we have used the monotonicity of Fi, i = 1, 2. Byinduction, this shows that both sequences (v1,n (t, x))n≥0 and (v2,n (t, x))n≥0 are

increasing. Therefore the limits

v1 (t, x) := limn→∞

vn1 (t, x) and v2 (t, x) := limn→∞

vn2 (t, x)

exist for all t ∈ [0, τ∗) and x ∈ D. From the Monotone Convergence Theorem weconclude that

vi (t, x) = Fiv3−i (t, x) , i = 1, 2,

for all t ∈ [0, τ∗) and x ∈ D. Moreover,

vi (t, x) ≤ ϕ(t)B(t), i = 1, 2,

for all t ∈ [0, τ∗) and x ∈ D, and the result follows.

Corollary 3.4. Under the assumptions of Theorem 3.3, if

β

∫ ∞

0

(eAWr ∨ eBWr

)maxi=1,2

ϕβi (r)

dr ≤ 1,

then the mild solution of (3.1) is global.

4. Bounds for the Probability of Explosion in Finite Time

Throughout this section we make the following assumptions:

(1) β1 = β2 > 0,(2) the initial values in (1.3) are of the form

fi(x) = Liψ(x), x ∈ D, i = 1, 2,

where L1 and L2 are positive constants,(3) G(z) = z1+β1 , z ≥ 0.

As above we denote

A = mini=1,2

(1 + β1)κ3−i − κi , B = maxi=1,2

(1 + β1)κ3−i − κi ,

and assume that A > 0. We also abbreviate Λ := (κ1∧κ2)2

2 .


4.1. An upper bound for the probability of blowup in finite time. Con-sider the random variable τ∗∗ defined by

τ∗∗ := inf

t ≥ 0 :

∫ t

0

(eAWr ∨ eBWr

)e−Λβ1rdr ≥ 1

β1 ∥ψ∥β1

∞mini=1,2

1

Lβ1

i

.

It is easy to see that τ∗∗ ≤ τ∗. Furthermore, noticing that∫ t

0

(eAWr ∨ eBWr

)e−Λβ1rdr

=

∫ t

0

eAWr−Λβ1r1Wr<0dr +

∫ t

0

eBWr−Λβ1r1Wr≥0dr

≤∫ ∞

0

e−Λβ1rdr +

∫ t

0

eBWr−Λβ1rdr =1

Λβ1+

∫ t

0

eBWr−Λβ1rdr,

it follows that

τ′′ := inf

t ≥ 0 :

1

Λβ1+

∫ t

0

eBWr−Λβ1rdr ≥ 1

β1 ∥ψ∥β1

∞mini=1,2

1

Lβ1

i

(4.1)

satisfies τ′′ ≤ τ∗∗ as long as A > 0.

Theorem 4.1. Assume that

∥ψ∥β1

∞Λ

< mini=1,2

1

Lβ1

i

.

Then

P (τ <∞) ≤

γ

2Λβ1

B2 , 2

B2

(1

β1∥ψ∥β1∞mini=1,2

1

Lβ1i

− 1

Λβ1

)

Γ(

2Λβ1

B2

) . (4.2)

Proof. From the relation τ′′ ≤ τ and the continuity of paths of Brownian motion,it follows that

P (τ <∞) ≤ P (τ′′ <∞)

= 1− P

(∫ ∞

0

eBWr−Λβ1rdr ≤ 1

β1 ∥ψ∥β1

∞mini=1,2

1

Lβ1

i

− 1

Λβ1

)

= P

(∫ ∞

0

eBWr−Λβ1rdr >1

β1 ∥ψ∥β1

∞mini=1,2

1

Lβ1

i

− 1

Λβ1

).

The result follows from (2.1).

Remark 4.2. Notice that P (τ <∞) < δ for any given δ > 0 provided that thepositive constants L1, L2 are sufficiently small, i.e., for sufficiently small initialconditions, the system (1.3) explodes in finite time with small probability.


4.2. Lower bound for the probability of explosion in finite time.

Theorem 4.3. If m = λ+ 12 (κ1 ∨ κ2)

2then

P (τ <∞) ≥ 8mβ1A2

∑n≥1

exp

− A22β1

8β1(L1+L2)β1∥ψ∥2β1

2

j22mβ1A2 −1,n

j22mβ1

A2 −1,n

. (4.3)

Proof. From the relation τ ≤ τ ′, the continuity of paths of Brownian motion andTheorem 2.6, it follows that

P (τ <∞) ≥ P (τ ′ <∞)

= P

(∫ ∞

0

e−(AWs−mβ1s)1AWs−mβ1sds ≥2β1

β1 (L1 + L2)β1 ∥ψ∥2β1

2

)

=

∫ ∞

2β1

β1(L1+L2)β1∥ψ∥2β12

mβ1∑n≥1

exp

−(A2

8j22mβ1

A2 −1,n

)y

dy.

=8mβ1A2

∑n≥1

exp

− A22β1

8β1(L1+L2)β1∥ψ∥2β1

2

j22mβ1A2 −1,n

j22mβ1

A2 −1,n

,

where we used the Monotone Convergence Theorem to obtain the last equality.

Remark 4.4. Notice that for sufficiently large L1 and L2, the relation

8mβ1A2

∑n≥1

exp

− A22β1

8β1(L1+L2)β1∥ψ∥2β1

2

j22mβ1A2 −1,n

j22mβ1

A2 −1,n

∼ 1−√8mβ

1/21 2β1/2

Aπ (L1 + L2)β1/2 ∥ψ∥β1

2

holds; see [4, formula (39)]. Therefore P (τ <∞) > 1 − ϵ for any given ϵ > 0provided that the positive constants L1, L2 are sufficiently large, i.e., for sufficientlylarge initial conditions, the solution of system (1.3) explodes in finite time withhigh probability.

References

1. Birkner, M., Lopez-Mimbela, J. A., and Wakolbinger, A.: Blow-up of semilinear PDE’s atthe critical dimension. A probabilistic approach, Proc. Amer. Math. Soc. 130 (2002), no. 8,

2431–2442.2. Blumenthal, R. M. and Getoor, R. K.: Markov Process and Potential Theory, Academic

Press, New York and London, 1968.3. Borodin, A. N. and Salminen, P.: Handbook of Brownian Motion–Facts and Formule, 2nd

edition, Springer, Basel, 2002.4. Colombaro, I., Giusti, A. and Mainardi, F.: A class of linear viscoelastic models based on

Bessel functions, Meccanica 52 (2017), no. 4-5, 825–832.

5. Doss, H.: Liens entre equations differentielles stochastiques et ordinaires, Annales de l’I.H.P.Probabilites et statistiques 13 (1977), no. 2, 99–125.

6. Dozzi, M., Kolkovska, E.T. and Lopez-Mimbela, J. A.: Exponential functionals of Brownianmotion and explosion times of a system of semilinear SPDEs, Stoch. Anal. Appl. 31 (2013),

no. 6, 975–991.


7. Dozzi, M. and Lopez-Mimbela, J. A.: Finite-time blowup and existence of global positive

solutions of a semi-linear SPDE, Stoch. Proc. Appl. 120 (2010), no. 6, 767–776.8. Dufresne, D.: The distribution of a perpetuity, with applications to risk theory and pension

funding, Scand. Acturial J. 1990 (1990), no. 1-2, 39–79.9. Engelbert, H. J. and Senf, T.: On functionals of a Wiener process with drift and exponential

local martingales, in Stochastic processes and related topics (Georgenthal, 1990), Math. Res.61, Akademie-Verlag, Berlin (1991), 45–58.

10. Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F.G.: Higher TranscendentalFunctions. Vol. II. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981.

11. Lopez-Mimbela, J. A. and Perez, A.: Global and nonglobal solutions of a system of nonau-tonomous semilinear equations with ultracontractive Levy generators, J. Math. Anal. Appl.423 (2015), no. 1, 720–733.

12. Pintoux, C. and Privault, N.: A direct solution to the Fokker-Planck equation for exponential

Brownian functionals, Anal. Appl. (Singap.) 8 (2010), no. 3, 287–304.13. Salminen, P. and Yor, M.: Perpetual integral functionals as hitting and occupation times,

Electron. J. Probab. 10 (2005), no. 11, 371–419.14. Sato, K. T.: Levy Proceses and Infinitely Divisible Distributions, Cambridge University

Press, 1999.15. Skorohod, A. V.: Random Processes with Independent Increments, Kluwer Academic Pub-

lishers, 1986.

16. Teschl, G.: Ordinary Differential Equations and Dynamical Systems, American Mathemat-ical Society, 2012.

17. Yor, M.: Exponential Functionals of Brownian Motion and Related Processes, Springer-Verlag, Berlin, 2001.

Eugenio Guerrero: Centro de Investigacion en Matematicas, Guanajuato, Guana-

juato 36240, MexicoE-mail address: [email protected]

Jose Alfredo Lopez-Mimbela: Centro de Investigacion en Matematicas, Guanaju-

ato, Guanajuato 36240, MexicoE-mail address: [email protected]

ONE DIMENSIONAL COMPLEX ORNSTEIN-UHLENBECK

OPERATOR

YONG CHEN*

Abstract. We show that for any fixed θ ∈ (−π2, 0) ∪ (0, π

2), the complex

Ornstein-Uhlenbeck operator

Lθ = 4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z,

is a normal but nonsymmetric diffusion operator.

1. Introduction

In [16], the authors show that for a continuos-time stationary L2-exponentialergodic Markov process, if its infinitesimal generator A is a normal but non-symmetric operator then all its power spectrums of real-valued observables aremonotonic on [0,∞) if and only if the eigenvalues of A satisfy the so-called IR-ratio rule:

|Imz||Rez|

≤ 1√3, ∀z ∈ σ(A) \ 0 .

Shortly after that, the authors apply the above IR-ratio rule to several finite statenormal but non-symmetric Markov operators in [7].

However, few normal but non-symmetric diffusion operators are well-studiedin the literature. For example, the following 2-dimensional Ornstein-Uhlenbeckprocess [

dX1(t)dX2(t)

]=

[−λ −ωω −λ

] [X1(t)X2(t)

]dt+

√2a

[dB1

t

dB2t

](1.1)

is used to model the Chandler wobble, i.e., the variation of latitude concerningwith the rotation of the earth, by M. Arato, A.N. Kolmogorov and Ya.G. Sinai [3](see also [1, 2]) and to model the motion of a charged test particle in the presenceof a constant magnetic field in [4, pp. 181–186]. But as far as we know, manymathematical properties such as the normality of the generator of (1.1) are stillnot written down in the previous literature.

Received 2017-6-25; Communicated by A. Boukas.

2010 Mathematics Subject Classification. Primary 60H10, Secondary, 60H07, 60G15.Key words and phrases. Ornstein-Uhlenbeck semigroup, complex Hermite polynomials, nor-

mal operator, normal diffusion operator.

* This work was supported by China Scholarship Council (201608430079).

357



358 YONG CHEN

For simplicity, we can assume that λ+ iω = eiθ, a = cos θ and then rewrite thegenerator as a complex-valued operator:

Lθ = 4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z, (1.2)

where θ ∈ (−π2 ,

π2 ) is fixed and ∂f

∂z = 12 (

∂f∂x − i∂f∂y ),

∂f∂z = 1

2 (∂f∂x + i∂f∂y ) are the

Wirtinger derivatives of f at point z = x+iy with x, y ∈ R. In [8], it is shown thatthe eigenfunctions are the complex Hermite polynomials and form an orthonormal

basis of L2(γ) where dγ = 12π e

− x2+y2

2 dxdy (see Proposition 2.2 below).

In this paper, we will firstly show that Lθ can be realized as an unboundednormal operator (see [19, p. 368]) in L2(γ) but nonsymmetric when θ = 0. Sec-ondly, we extend the known fact about the 1-dimensional real symmetric diffusionoperator [6, 22, 23] to the complex case. Precisely stated, we present the explicit

expression of Lθ in L2(γ) (see Theorem 4.3) and show that it is a normal diffusionoperator (see Theorem 4.4).

This article is organized as follows. Section 2 provides necessary informationof complex Hermite polynomials. Section 3 contains the proof of the normality ofthe complex Ornstein-Uhlenbeck semigroup. Section 4 contains the main resultson the explicit expression of Lθ and the property of the normal diffusion operator.Finally, some necessary approximation of identity and N-representation theoremare listed in Appendix.

2. Preliminaries

Definition 2.1. (Definition of the complex Hermite polynomials [8, Def. 2.4]) Wecall ∂ := ∂

∂z and ∂ := ∂∂z the complex annihilation operators. Let m,n ∈ N. We

define the sequence on C (or say: R2)

J0,0(z) = 1,

Jm,n(z) =

√2m+n

m!n!(∂∗)m(∂∗)n1,

where (∂∗ϕ)(z) = − ∂∂zϕ(z) +

z2ϕ(z), (∂

∗ϕ)(z) = − ∂∂zϕ(z) +

z2ϕ(z) for ϕ ∈ C1

↑(R2)

(see Definition 5.1) are the adjoint of the operators ∂, ∂ in L2(γ) respectively (thecomplex creation operator).

In [8, Theorem 2.7, Corollary 2.8], the authors show that Jm,n(z) satisfies that:

Proposition 2.2. The complex Hermite polynomials Jm,n(z) : m,n ∈ N form

an orthonormal basis of L2(γ) where dγ = 12π e

− x2+y2

2 dxdy. Thus, every function

f in L2(γ) has a unique series expression

f =∞∑

m=0

∞∑n=0

bm,nJm,n(z),

where the coefficients bm,n are given by

bm,n = ⟨f, Jm,n⟩ =∫R2

fJm,n(z)dγ.

COMPLEX ORNSTEIN-UHLENBECK OPERATOR 359

Moreover, for any θ ∈ (−π2 ,

π2 ) and each m,n ∈ N,

LθJm,n(z) = −[(m+ n) cos θ + i(m− n) sin θ]Jm,n(z). (2.1)

The real Hermite polynomials are defined by the formula1

Hn(x) =(−1)n√n!

ex2/2 dn

dxne−x2/2, n = 1, 2, . . . .

The following property gives the fundamental relation between the real and thecomplex Hermite polynomials [8, Corollary 2.8].

Proposition 2.3. Let z = x + iy with x, y ∈ R. Both Jk,l(z) : k + l = n andHk(x)Hl(y) : k + l = n generate the same linear subspace of L2(γ).

3. The Normality of the Complex Ornstein-Uhlenbeck Semigroup

The Ornstein-Uhlenbeck process (1.1) can be rewritten as a complex-valuedprocess:

dZt = −eiθZtdt+√2 cos θdζt, t ≥ 0,

Z0 = x ∈ C, (3.1)

where θ ∈ (−π2 ,

π2 ), and ζt = B1(t)+iB2(t) is a complex Brownian motion. Solving

for Z gives

Zt = e−(cos θ+i sin θ)t(x+

√2 cos θ

∫ t

0

e(cos θ+i sin θ)s dζs). (3.2)

Thus, the associated Ornstein-Uhlenbeck semigroup of Eq.(3.1) has the followingexplicit representation, due to Kolmogorov, for each φ ∈ Cb(R2) (the space of allcontinuous and bounded complex-valued functions on R2),

Ptφ(x) = Ex[φ(Zt)] (3.3)

=1

2π(1− e−2t cos θ)

∫R2

e− |y|2

2(1−e−2t cos θ)φ(e−(cos θ+i sin θ)tx− y) dy1dy2,

(3.4)

where y = y1 + iy2 and x, y ∈ C and we write a function φ(y1, y2) of the two realvariables y1 and y2 as φ(y) of the complex argument y1 + iy2 (i.e., we use thecomplex representation of R2 in (3.3-3.4)). The change of variable formula yieldsthe following Mehler formula [8, p. 584].

Proposition 3.1. (Mehler formula) For each φ ∈ Cb(R2),

Ptφ(x) =

∫Cφ(e−(cos θ+i sin θ)tx+

√1− e−2t cos θy) dγ(y), (3.5)

where

dγ(y) =1

2πexp

− (y21 + y22)

2

dy1dy2. (3.6)

1Note that Hn(x) =(−1)n

n!ex

2/2 dn

dxn e−x2/2 in [14, 20] and Hn(x) = (−1)nex2/2 dn

dxn e−x2/2

in [8, 11], here we use the definition in [23].

360 YONG CHEN

Similar to the real Ornstein-Uhlenbeck semigroup [20, Proposition 2.3], usingthe rotation invariant of the measure γ and Lebesgue’s dominated convergencetheorem, it follows from Proposition 3.1 that γ is the unique invariant measure ofPt. In detail, for each φ ∈ Cb(R2),∫

CPtφ(x) dγ(x) =

∫Cφ(x) dγ(x) (3.7)

and

limt→∞

Ptφ(x) =

∫Cφ(y) dγ(y), ∀x ∈ C. (3.8)

Denote the associated transition probabilities on C as Pt(x,A) = Pt1A(x) foreach A ∈ B(R2). Along the same line of the real case [6, 20, 23], for each p ≥ 1, itfollows from Jensen’s inequality that for each φ ∈ Cb(R2),

∥Ptφ∥pLp(γ) =

∫C|Ptφ(x)|p dγ(x) =

∫C

∣∣∣∣∫Cφ(y)Pt(x, d y)

∣∣∣∣p dγ(x)

≤∫Cdγ(x)

∫C|φ(y)|p Pt(x, d y)

=

∫C|φ|p (x) dγ(x) (by (3.7))

= ∥φ∥pLp(γ) .

It follows from the B.L.T. theorem [17, p. 9] that Ptt≥0 can be uniquely extended

to a strong continuous contraction semigroup T pt t≥0 on Lp(γ) for each p ≥ 1

2. Let Ap be the (infinitesimal) generator, then Ap is closed and D(Ap) = Lp(γ)(i.e., densely defined) [15, 17].

Lemma 3.2. Suppose Y = (y1, . . . , yn), Z = (z1, . . . , zn) ∈ Cn and Y = MZ,where M = (Mij) is an n-by-n unitary matrix over the field C. If zi, i = 1 . . . , n

are independent, each being centered complex normal such that E |zi|2 = σ2, thenyi, i = 1 . . . , n are also independent, each being centered complex normal such thatE |yi|2 = σ2.

Proof. It follows from [9, Theorem 1.1] that yi, i = 1 . . . , n are centered complexnormal. In addition, we have that

E[yiyj ] =∑k,l

MikE[zkzl]Mjl = σ2∑k

MikMjk = σ2δij ,

where δij is the Kronecker delta. Thus, yi, i = 1 . . . , n have independent identicaldistributions with variance σ2. Proposition 3.3. For each θ ∈ (−π

2 ,π2 ), denote the semigroup Pt depending on

θ in (3.3) by P θt , then for each ϕ ∈ Cb(R2)

P θt (P

θt )

∗ϕ(x) = (P θt )

∗P θt ϕ(x), (3.9)

where (P θt )

∗ is the adjoint operator of P θt in L2(γ). Furthermore, when restricted

on Cb(R2), (P θt )

∗ = P−θt .

2Namely, T pt is the closure (see [17, p. 250]) in Lp(γ) of the operator Pt.


Proof. Set α = eiθ. For each ϕ, ψ ∈ Cb(R2) and t ≥ 0, we have that

⟨P θt ϕ, ψ⟩ =

∫Cψ(z1) dγ(z1)

∫Cϕ(e−αtz1 +

√1− e−2tReαz2) dγ(z2)

=

∫Cϕ(y1) dγ(y1)

∫Cψ(e−αty1 −

√1− e−2tReαy2) dγ(y2) (3.10)

=

∫Cϕ(y1) dγ(y1)

∫Cψ(e−αty1 +

√1− e−2tReαy2) dγ(y2) (3.11)

= ⟨ϕ, P−θt ψ⟩,

where (3.10) is deduced from Lemma 3.2 by taking n = 2 and

M =

[e−αt

√1− e−2tReα

−√1− e−2tReα e−αt

],

and (3.11) is deduced from the rotation invariant of the measure γ. Therefore,

the adjoint operator of P θt in L2(γ) satisfies that (P θ

t )∗ = P−θ

t when restricted onCb(R2) for each θ ∈ (−π

2 ,π2 ). Thus, for each ϕ ∈ Cb(R2),

P θt (P

θt )

∗ϕ(x)

=

∫C2

ϕ(e−αt(e−αtx+

√1− e−2tReαz1) +

√1− e−2tReαz2

)dγ(z1)dγ(z2)

=

∫Cϕ(e−2tReαx+

√1− e−4tReαz

)dγ(z) (3.12)

=

∫C2

ϕ(e−αt(e−αtx+

√1− e−2tReαz1) +

√1− e−2tReαz2

)dγ(z1)dγ(z2)

= (P θt )

∗P θt ϕ(x),

where (3.12) is deduced from the well-known fact that if Z1, Z2 are two inde-

pendent standard complex normal random variables, then e−αt√1− e−2tReαZ1 +√

1− e−2tReαZ2 and√1− e−4tReαZ1 have the same law [9, Theorem 1.1].

Theorem 3.4.T 2t

t≥0

is a semigroup of normal operators (see [19, p. 382]) in

L2(γ) and thus the generator A2 is a normal operator in L2(γ).

Proof. Since T 2t is the closure of the contraction operator Pt in L2(γ), it follows

from (c) of Theorem VIII.1 in [17, p. 253] that the adjoint operator of T 2t equals

to that of Pt. It follows from the density argument that (3.9) can be extended toeach ϕ ∈ L2(γ), i.e.,

T 2t (T

2t )

∗ϕ = (T 2t )

∗T 2t ϕ, ∀ϕ ∈ L2(γ).

ThusT 2t

t≥0

is a semigroup of normal operators. It follows from [19, Theo-

rem 13.38] that the generator A2 is a normal operator in L2(γ).

4. The Normal Diffusion Operators in C

The first aim of this section is to show the explicit expression of the generatorA2.

362 YONG CHEN

Definition 4.1. For any θ ∈ (−π2 ,

π2 ), define

D(Lθ) =

f ∈ L2(γ),

∞∑m=0

∞∑n=0

(m2 + n2 + 2mn cos 2θ) |⟨f, Jm,n⟩|2 <∞

(4.1)

and

Lθf = −∞∑

m=0

∞∑n=0

[(m+ n) cos θ + i(m− n) sin θ]⟨f, Jm,n⟩Jm,n(z). (4.2)

Theorem 4.2. If ϕ ∈ C2↑(R2), then ϕ ∈ D(Lθ) and

Lθϕ = [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z]ϕ. (4.3)

Theorem 4.3. Let A2 be as in Theorem 3.4 and Lθ be as in Definition 4.1. Forany θ ∈ (−π

2 ,π2 ), A2 = Lθ, i.e., D(A2) = D(Lθ) and A2φ = Lθφ on D(Lθ).

The second aim of this section is to show that the operator Lθ defined abovesatisfies the following theorem, which is named as the normal diffusion operatoranalogous to the symmetric diffusion operator given by Stroock [6, 22, 23].

Theorem 4.4. The densely defined linear closed operator Lθ defined in Proposi-tion 4.1 is a normal diffusion operator. Namely, it satisfies that:

1) Lθ is a normal operator on D(Lθ).2) 1 ∈ D(Lθ) and Lθ1 = 0.3) There exists a linear subspace

D ⊂ϕ ∈ D(Lθ) ∩ L4(γ) : Lθϕ ∈ L4(γ), |ϕ|2 ∈ D(Lθ)

such that graph(Lθ|D) is dense in graph(Lθ).

4) For any θ ∈ (−π2 ,

π2 ), define(

ϕ, ψ)θ=

1

2 cos θ[Lθ(ϕψ)− ϕLθ(ψ)− ψLθ(ϕ)]

for ϕ, ψ ∈ D. Then(·, ·

)θ: D × D → L2(γ) is a non-negative definite

bilinear form on the field C.5) (Diffusion property)If ϕ = (ϕ1, . . . , ϕn) ∈ Dn and F ∈ C2

↑(Cn), then F ϕ ∈ D(Lθ) and

Lθ(F ϕ)

= cos θn∑

i,j=1

(ϕi, ϕj

)θ

∂2F

∂zi∂zj ϕ+

(ϕi, ϕj

)θ

∂2F

∂zi∂zj ϕ+ 2

(ϕi, ϕj

)θ

∂2F

∂zi∂zj ϕ

+

n∑i=1

Lθϕi∂F

∂zi ϕ+ Lθϕi

∂F

∂zi ϕ. (4.4)

6) Lθ has an extension A1 to L1(γ) with domain D(A1) such that

D(Lθ) =ϕ ∈ D(A1) ∩ L2(γ) : A1ϕ ∈ L2(γ)

,

i.e., the closure of Lθ in L1(γ) is A1.

Proofs of Theorems 4.2-4.4 are presented in Section 4.1.


4.1. Proofs of Theorems.

Proposition 4.5. Let Lθ be as in Definition 4.1. Then Lθ is closed on L2(γ).

Proof. Suppose that fk ∈ D(Lθ) such that fk → f, Lθfk → g in L2(γ), we willshow that f ∈ D(Lθ) and Lθf = g. In fact, by Fatou’s lemma and Parseval’sidentity, we have that

∞∑m,n=0

(m2 + n2 + 2mn cos 2θ) |⟨f, Jm,n⟩|2

≤ lim infk→∞

∞∑m,n=0

(m2 + n2 + 2mn cos 2θ) |⟨fk, Jm,n⟩|2

= lim infk→∞

∥Lθfk∥2

= ∥g∥2 <∞.

Thus f ∈ D(Lθ). For each m,n ≥ 0, we have that

limk→∞

⟨[(m+ n) cos θ + i(m− n) sin θ]fk + g, Jm,n⟩

= ⟨[(m+ n) cos θ + i(m− n) sin θ]f + g, Jm,n⟩It follows from Parseval’s identity and Fatou’s lemma that

∥Lθf − g∥2 =∞∑

m,n=0

|⟨[(m+ n) cos θ + i(m− n) sin θ]f + g, Jm,n⟩|2

≤ lim infk→∞

∞∑m,n=0

|⟨[(m+ n) cos θ + i(m− n) sin θ]fk + g, Jm,n⟩|2

= lim infk→∞

∥Lθfk − g∥2 = 0.

Thus Lθf = g. Remark 4.6. Suppose that Hm,n(x, y) = Hm(x)Hn(y) is the Hermite polynomialof two variables. Then it follows from Proposition 2.3 that∑

m+n=l

|⟨f, Jm,n⟩|2 =∑

m+n=l

|⟨f, Hm,n⟩|2 .

Together with

(m+n)2 ≥ m2+n2+2mn cos 2θ = (m+n)2 cos θ2+(m−n)2 sin θ2 ≥ (m+n)2 cos θ2,

we deduce that

D(Lθ) =

f ∈ L2(γ),

∞∑m=0

∞∑n=0

(m+ n)2 |⟨f, Jm,n⟩|2 <∞

=

f ∈ L2(γ),

∞∑m=0

∞∑n=0

(m+ n)2 |⟨f, Hm,n⟩|2 <∞

, (4.5)

that is to say, D(Lθ) is independent to θ. In fact, the right hand side of (4.5) isexact the Sobolev weighted space H2

γ , please refer to [12] for details.

364 YONG CHEN

Proposition 4.7. For any θ ∈ (−π2 ,

π2 ), Lθ is a normal operator on D(Lθ) such

that 1 ∈ D(Lθ) and Lθ1 = 0.

Proof. Suppose that f, g ∈ D(Lθ). It follows from Parseval’s identity that

⟨Lθf, g⟩ = −∞∑m,n

[(m+ n) cos θ + i(m− n) sin θ]⟨f, Jm,n⟩⟨g, Jm,n⟩

= −∞∑m,n

⟨f, Jm,n⟩[(m+ n) cos θ − i(m− n) sin θ]⟨g, Jm,n⟩

= ⟨f, L−θg⟩.

Thus, the adjoint operator of Lθ is L∗θ = L−θ. The equality (4.5) implies that

D(Lθ) = D(L−θ) = D(L∗θ).

And for each f ∈ D(Lθ) such that Lθf ∈ D(L∗θ), we have that

L∗θLθf =

∑m,n

(m2 + n2 + 2mn cos 2θ)⟨f, Jm,n⟩Jm,n(z) = LθL∗θf.

Therefore, Lθ is a normal operator on D(Lθ). 1 ∈ D(Lθ) and Lθ1 = 0 is trivial.

Proposition 4.8. Denote by D = span Jm,n, m, n ≥ 0 the linear span (alsocalled the linear hull) of complex Hermite polynomials. Then

D ⊂ϕ ∈ D(Lθ) ∩ L4(γ) : Lθϕ ∈ L4(γ), |ϕ|2 ∈ D(Lθ)

(4.6)

and graph(Lθ|D) is dense in graph(Lθ).

Proof. The equality [8, Theorem 2.5]

Jm,n(z) = (m!n!2m+n)−12

m∧n∑r=0

(−1)r2rm!n!

(m− r)!(n− r)!r!zm−r zn−r, ∀m,n ≥ N

and the equality [8, Corollary 2.8]

zmzn =m∧n∑k=0

m!n!

(m− k)!(n− k)!k!

√(m− k)!(n− k)!2m+nJm−k,n−k(z), ∀m,n ≥ N

imply that D = span Jm,n, m, n ≥ 0 = span zmzn, m, n ≥ 0. But f(z) =zmzn belonging to the right hand side of (4.6) is trivial. Thus (4.6) holds.

Since D is a dense subset of L2(γ) (see Proposition 2.2), D is dense in D(Lθ).Note that Lθ is a closed operator, we get that graph(Lθ|D) is dense in graph(Lθ).

Proof of Theorem 4.2. First, it follows from Proposition 2.2 that (4.3) holds whenϕ ∈ D = span Jm,n, m, n ≥ 0.

Second, suppose that ϕ ∈ L2(γ) satisfies that the sequence

am,n = ⟨ϕ, Hm(x)Hn(y)⟩


is rapidly decreasing, then we will show that ϕ ∈ D(Lθ) and (4.3) is satis-fied. In fact, it follows from Proposition 5.8 that the Hermite expansion ϕ(z) =∑∞

m,n=0 am,nHm(x)Hn(y) satisfies that∥∥∥∥xk1yk2∂p1+p2

∂xp1∂yp2(ϕ− ϕl)

∥∥∥∥L2(γ)

→ 0 as l → ∞, ∀k1, k2, p1, p2 ∈ N,

where ϕl =∑

m+n≤l am,nHm(x)Hn(y). Thus,∥∥∥∥zk1 zk2∂p1+p2

∂zp1∂zp2(ϕ− ϕl)

∥∥∥∥L2(γ)

→ 0 as l → ∞, ∀k1, k2, p1, p2 ∈ N.

It follows from Proposition 2.3 that∑m+n≤l

am,nHm(x)Hn(y) =∑

m+n≤l

bm,nJm,n(z).

Thus, as l → ∞, we have that in L2(γ), ϕl → ϕ and

Lθϕl = −∑

m+n≤l

[(m+ n) cos θ + i(m− n) sin θ]bm,nJm,n(z)

= [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z]ϕl

→ [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z]ϕ.

Since Lθ is closed, we have that ϕ ∈ D(Lθ) and (4.3) is satisfied.Finally, it follows from Proposition 5.5 that if ϕ ∈ C2

↑(R2) then there exists

an approximation of identity Bϵϕ ∈ C∞c (R2) such that for all p1 + p2 ≤ 2 and

k1, k2 ≥ 0, xk1yk2 ∂p1+p2

∂xp1∂yp2(Bϵϕ) → xk1yk2 ∂p1+p2

∂xp1∂yp2ϕ in L2(γ) as ϵ → 0. In ad-

dition, it follows from Proposition 5.6 that the sequence ⟨Bϵϕ, Hm(x)Hn(y)⟩ israpidly decreasing. Thus, as ϵ→ 0, we have that in L2(γ), Bϵϕ→ ϕ and

Lθ(Bϵϕ) = [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z]Bϵϕ

→ [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z]ϕ.

Since Lθ is closed, we have that ϕ ∈ D(Lθ) and (4.3) is satisfied. Proof of Theorem 4.3. First, it follows from the density argument (see Proposi-tion 5.5) and Lebesgue’s dominated convergence theorem that the Mehler formula(3.5) is still valid for φ ∈ C0

↑(R2), i.e.,

T 2t φ(x) =

∫Cφ(e−(cos θ+i sin θ)tx+

√1− e−2t cos θy) dγ(y), ∀φ ∈ C0

↑(R2).

Then T 2t φ(x) = Ptφ(x) = Ex[φ(Zt)] for each φ ∈ C0

↑(R2).

366 YONG CHEN

Second, using (3.1), it follows from Ito’s lemma and Theorem 4.2 that for eachφ ∈ C2

↑(R2),

φ(Zt) = φ(x) +

∫ t

0

∂

∂zφ(Zs) dZs +

∫ t

0

∂

∂zφ(Zs) dZs +

∫ t

0

∂2

∂z∂z(Zs)d⟨Z, Z⟩s

= φ(x) +

∫ t

0

Lθφ(Zs) ds−√2 cos θ(

∫ t

0

∂φ

∂z(Zs)dζs +

∫ t

0

∂φ

∂z(Zs)dζs).

Then

T 2t φ(x) = Ex[φ(Zt)] = φ(x) + Ex[

∫ t

0

Lθφ(Zs) ds]

= φ(x) +

∫ t

0

Ex[Lθφ(Zs)] ds (by Fubini Theorem)

= φ(x) +

∫ t

0

T 2s Lθφ(x) ds,

and

A2φ = limt↓0

T 2t φ− φ

t= lim

t↓0

1

t

∫ t

0

T 2s Lθφ(x) ds

= Lθφ (in L2(γ)),

where to get the last equality we use the continuity of t→ T 2t φ for any φ ∈ L2(γ)

(see [15, Corollary 2.3] or part (a) of [15, Theorem 2.4]). Therefore, A2 = Lθ onC2

↑(R2),

Third, since graph(Lθ|C2↑(R2)) is dense in graph(Lθ) (see Proposition 4.8) and

A2 is closed, we have that Lθ ⊆ A2. It follows from Proposition 4.7 and Theo-rem 3.4 that both Lθ and A2 are normal operators. Since Lθ is maximally normal(see [19, Theorem 13.32]), we have that A2 = Lθ. Corollary 4.9. For any θ ∈ (−π

2 ,π2 ), Lθ ⊆ A1 (i.e., A1 is an extension of Lθ to

L1(γ)) and

D(Lθ) =ϕ ∈ D(A1) ∩ L2(γ) : A1ϕ ∈ L2(γ)

. (4.7)

Proof. The proof is similar to the real case [6, p. 19]. In detail,

D(Lθ) = D(A2) ⊂ϕ ∈ D(A1) ∩ L2(γ) : A1ϕ ∈ L2(γ)

is trivial. Now suppose that ϕ ∈ D(A1) ∩ L2(γ) and A1ϕ ∈ L2(γ), then as t→ 0,

T 2t ϕ− ϕ

t=T 1t ϕ− ϕ

t

=1

t

∫ t

0

T 1sA1ϕds (by the semigroup equation)

=1

t

∫ t

0

T 2sA1ϕds→ A1ϕ (in L2(γ)),

where we use again the continuity of t → T 2t φ for the last equality. Thus ϕ ∈

D(A2) = D(Lθ) and (4.7) holds.


Proof of Theorem 4.4. By Proposition 4.5, Lθ is closed. 1)-3) and 6) of Theo-rem 4.4 are shown by Proposition 4.7-4.8 and Corollary 4.9 respectively. Since

F ∈ C2↑(Cn) and ϕ = (ϕ1, . . . , ϕn) ∈ Dn, then F ϕ ∈ C2

↑(R2). By the complex

version of the chain rule [21, p. 27], it follows from Theorem 4.2 that

Lθ(F ϕ) = [4 cos θ∂2

∂z∂z− eiθz

∂

∂z− e−iθ z

∂

∂z](F ϕ)

= −eiθzn∑

i=1

∂

∂zϕi∂F

∂zi ϕ+

∂

∂zϕi∂F

∂zi ϕ

− e−iθ zn∑

i=1

∂

∂zϕi∂F

∂zi ϕ+

∂

∂zϕi∂F

∂zi ϕ

+ 4 cos θn∑

i=1

∂2ϕi∂z∂z

∂F

∂zi ϕ+

∂2ϕi∂z∂z

∂F

∂zi ϕ

+ 4 cos θn∑

i,j=1

∂

∂zϕi(

∂

∂zϕj

∂2F

∂zi∂zj ϕ+

∂

∂zϕj

∂2F

∂zi∂zj ϕ)

+ 4 cos θ

n∑i,j=1

∂

∂zϕi(

∂

∂zϕj

∂2F

∂zi∂zj ϕ+

∂

∂zϕj

∂2F

∂zi∂zj ϕ)

= 4 cos θn∑

i,j=1

∂

∂zϕi

∂

∂zϕj

∂2F

∂zi∂zj ϕ+

∂

∂zϕi

∂

∂zϕj

∂2F

∂zi∂zj ϕ

+ 4 cos θn∑

i,j=1

(∂

∂zϕi

∂

∂zϕj +

∂

∂zϕi

∂

∂zϕj)

∂2F

∂zi∂zj ϕ

+

n∑i=1

Lθϕi∂F

∂zi ϕ+ Lθϕi

∂F

∂zi ϕ. (4.8)

Taking F (z1, z2) = z1z2 in the above equation, we have that(ϕ, ψ

)θ=

1

2 cos θ[Lθ(ϕψ)− ψLθ(ϕ)− ϕLθ(ψ)]

= 2[∂ϕ

∂z

∂ψ

∂z+∂ϕ

∂z

∂ψ

∂z] = 2[

∂ϕ

∂z

∂ψ

∂z+∂ϕ

∂z

∂ψ

∂z]. (4.9)

Hence,(ϕ, ψ

)θis a non-negative definite bilinear form on the field C. Substituting

(4.9) into (4.8), we show (4.4). Thus, 4)-5) of Theorem 4.4 are obtained.

5. Appendix

To be self-contained, we list the necessary results of functions slowly increasingat infinity. Some results which can not be found in textbooks will be shown shortlyhere. In this section all functions will be complex-valued and defined on Rn.

Definition 5.1. Denote by C∞c (Rn) the space of smooth and compactly supported

functions on Rn [5, p. 5]). Denote by S(Rn) the space of C∞ functions rapidlydecreasing at infinity [5, p. 105]. We say that a continuous function f(x) is slowly

368 YONG CHEN

increasing at infinity if there exists an integer k such that (1+r2)−k2 f(x) is bounded

in Rn with r = |x| [5, p. 110]. Denote by Cm↑ (Rn) the space of all functions having

slowly increasing at infinity continuous partial derivatives of order ≤ m.

Notation 1. Denote by γ the n-dimensional standard Gaussian measure:

dγ(x) = (2π)−n2 exp

−|x|2

2

dx, x ∈ Rn.

Denote the density function by ρ(x) = dγ(x)dx .

5.1. Approximation of identity of Cm↑ (Rn) in Lq(γ).

Notation 2. Set

ϕ(x) = e− 1

1−|x|2 1|x|<1, x ∈ Rn, (5.1)

where |x| =√x21 + · · ·+ x2n and 1B the characteristic function of set B. Divide

this function by its integral over the whole space to get a function α(x) of integralone which is called a mollifier. Next, for every ϵ > 0, define [5, p. 5]

αϵ(x) =1

ϵnα(x

ϵ). (5.2)

Let L1loc(Rn) be the space of locally integrable function on Rn. If u ∈ L1

loc(Rn),the function

uϵ(x) =

∫Rn

u(x− y)αϵ(y)dy =

∫Rn

αϵ(x− y)u(y)dy (5.3)

is said to be the convolution of u and αϵ [5, Definition 1.4]. It is also denoted bythe convolution operator Aϵu = (u ∗ αϵ)(x).

Lemma 5.2. Suppose that f(x) ∈ C0↑(Rn). Then Aϵf ∈ C∞

↑ (Rn) (the space of

C∞ functions slowly increasing at infinity) and for any q ≥ 1 and any k ∈ Nn,limϵ→0

xkAϵf = xkf in Lq(γ).

Proof. First, for any ϵ > 0, since αϵ ∈ C∞c (Rn) ⊂ S(Rn) and f(x) ∈ C0

↑(Rn) ⊂S′(Rn) (tempered distributions, see Example 4 in [5, 110]), it follows from Theo-rem 4.9 of [5, p. 133] that Aϵf = f ∗ αϵ ∈ C∞

↑ (Rn).

Second, since any polynomial P (x), x ∈ Rn, is in Lq(γ), we have f, Aϵf ∈Lq(γ). Thus,

∥Aϵf − f∥qq = limn→∞

∫Ba

|Aϵf − f |q dγ(x),

where Ba = x ∈ Rn, |x| ≤ a.Finally, given σ > 0, there exists Ba such that

∥Aϵf − f∥qq ≤∫Ba

|Aϵf − f |q dγ(x) +σ

2. (5.4)

Note that∫Ba

|Aϵf − f |q dγ(x) ≤ supx∈Ba

|Aϵf − f |q γ(Kn) ≤ supx∈Ba

|Aϵf − f |q . (5.5)


It follows from [5, Theorem 1.1] that Aϵf → f uniformly on Ba as ϵ → 0. Thus

there exists ϵ0 > 0 such that supx∈Ba|Aϵf − f | ≤ (σ2 )

1q for any 0 < ϵ < ϵ0.

Together with (5.4) and (5.5), we have that ∥Aϵf − f∥qq ≤ σ, which proves that

Aϵf → f in Lq(γ), as ϵ→ 0.Similar to the above proof, it follows that for any k ∈ Nn, lim

ϵ→0xkAϵf = xkf in

Lq(γ).

Corollary 5.3. Suppose that f(x) ∈ Cm↑ (Rn). Then for any p, k ∈ Nn such that

|p| ≤ m, xk∂p(Aϵf) → xk∂pf in Lq(γ), as ϵ→ 0.

Proof. First, if f(x) ∈ Cm↑ (Rn) then ∂pf ∈ C0

↑(Rn) for any p ∈ Nn such that

|p| ≤ m. It follows from Lemma 5.2 that xkAϵ(∂pf) → xk∂pf in Lq(γ) for any

k ∈ Nn. Second, for any p ∈ Nn, if u, ∂pu ∈ L1loc(Rn) then ∂p(Aϵu) = Aϵ(∂

pu).Finally, since C0

↑(Rn) ⊂ L1loc(Rn), we have that xk∂p(Aϵf) → xk∂pf in Lq(γ).

Notation 3. Let a ∈ R+ and denote by Ba+1 and Ba concentric balls of radiusa+1 and a, respectively. It follows from Corollary 3 of [5, p. 9] that there exists aso-called (smooth) cutoff function βa(x) ∈ C∞

c (Rn) such that: (i) 0 ≤ βa ≤ 1 andsuppβa ⊂ Ba+1, (ii) βa(x) = 1 on Ba, (iii) for all p ∈ Nn, supx∈R |∂pβa| ≤ c(n, p).

Lemma 5.4. Let the cutoff function βa prevail. Suppose that g ∈ C∞↑ (Rn) and

set ga = gβa. Then ga ∈ C∞c (Rn), and for any k, p ∈ Nn, lim

a→∞xk∂pga = xk∂pg

in Lq(γ) for any q ≥ 1.

Proof. The Lebniz’s rule implies that

∂pga =∑l≤p

p!

l!(p− l)!∂lβa∂

p−lg.

Denote Ga = Rn − Ba, it follows from (i)-(iii) of Notation 3 that

|∂pga − ∂pg| = 1Ga

∣∣∣∣∣∣(βa − 1)∂pg +∑

0<l≤p

p!

l!(p− l)!∂lβa∂

p−lg

∣∣∣∣∣∣≤ c× 1Ga

∑l≤p

∣∣∂p−lg∣∣ , (5.6)

where c = p! max0<l≤p

c(n, p− l)/l!(p− l)!.

Since g ∈ C∞↑ (Rn), we have h(x) := xk

∑l≤p

∣∣∂p−lg∣∣ ∈ C∞

↑ (Rn) ⊂ Lq(γ).

Therefore, h1Ga → 0 in Lq(γ) as a → ∞. Together with (5.6), we have thatlima→∞

xk∂pga = xk∂pg in Lq(γ).

Proposition 5.5. (Approximation of identity of Cm↑ (Rn) in Lq(γ))

Suppose that f(x) ∈ Cm↑ (Rn). Denote

Bϵf = β 1ϵ×Aϵf.

Then Bϵf ∈ C∞c (Rn), and for q ≥ 1 and k, p ∈ Nn such that |p| ≤ m, xk∂p(Bϵf) →

xk∂pf in Lq(γ), as ϵ→ 0.

370 YONG CHEN

Proof. Lemma 5.2 implies that Aϵf ∈ C∞↑ (Rn). Then it follows from Lemma 5.4

that Bϵf ∈ C∞c (Rn) and

∥∥xk∂p(Bϵf)− xk∂p(Aϵf)∥∥q→ 0. Corollary 5.3 implies

that ∥∥xk∂p(Aϵf)− xk∂pf∥∥q→ 0.

By the triangle inequality, we have that as ϵ→ 0,∥∥xk∂p(Bϵf)− xk∂pf∥∥q≤

∥∥xk∂p(Bϵf)− xk∂p(Aϵf)∥∥q+∥∥xk∂p(Aϵf)− xk∂pf

∥∥q

→ 0.

5.2. The N-representation theorem for S(Rn) in L2(γ). Suppose

Hl(x) =(−1)l√l!ex

2/2 dl

dxle−x2/2

is the l-th Hermite polynomial of one variable. It is well known that the set ofHermite polynomials of several variables

Hm :=

n∏k=1

Hmk(xk), m = (m1, . . . ,mn) ∈ Nn

(5.7)

is an orthonormal basis of L2(γ). Thus, every function u ∈ L2(γ) has a uniqueseries expression

u =∑

m∈Nn

amHm, (5.8)

where the coefficients am are given by

am =

∫Rn

u(x)Hm(x) dγ(x).

Proposition 5.6. u ∈ L2(γ) satisfies that am =∫Rn u(x)Hm(x) dγ(x) is rapidly

decreasing (i.e., for r ∈ Nn ≥ 0, am = O(m−r) as |m| → ∞) if and only if

u = fρ−12 with f ∈ S(Rn).

Proof. Denote the Hermite functions Hm(x) = Hm(x)ρ12 , then∫

Rn

u(x)Hm(x) dγ(x) =

∫Rn

f(x)Hm(x) dx.

The desired conclusion is followed from Theorem 3.5 and Exercise 3 of [10, p. 135].

Remark 5.7. Clearly, the smooth and compactly supported function satisfies theabove condition. In fact,

C∞c (Rn) =

u = fρ−

12 : f ∈ C∞

c (Rn)⊂

u = fρ−

12 : f ∈ S(Rn)

.

Proposition 5.8. If u ∈ L2(γ) satisfies that am =∫Rn u(x)Hm(x) dγ(x) is rapidly

decreasing, then the Hermite expansion u(x) =∑

m∈Nn amHm(x) satisfies that∥∥xk∂p(u− ul)∥∥L2(γ)

→ 0 as l → ∞, ∀k, p ∈ Nn,

where ul =∑

|m|≤l amHm(x).


Proof. Proposition 5.6 implies that S(Rn) ∋ f =∑

m amHm(x). Denote fl =

ulρ12 , then it follows from the N-representation theorem for S(Rn) (see Theo-

rem V.13 of [17, p. 143]) that fl → f in S(Rn) which means that: as l → ∞,∥∥xm∂i(f − fl)∥∥L2(dx)

→ 0 ∀m, i ∈ Nn.

The Lebniz’s rule implies that there exists a constant c > 0 such that∥∥xk∂p(u− ul)∥∥L2(γ)

≤ c∑

m≤k+p, i≤p

∥∥xm∂i(f − fl)∥∥L2(dx)

.

Thus the desired conclusion follows.

References

1. Arato, M.: Linear Stochastic Systems with Constant Coefficients. A Statistical Approach.

Lecture Notes in Control and Information Sciences, 45. Springer-Verlag, Berlin, 1982.2. Arato, M.; Baran, S. and Ispany M.: Functionals of complex Ornstein-Uhlenbeck processes.

Comput. Math. Appl. 37(1999), no. 1, 1–13.3. Arato, M.; Kolmogorov, A.N. and Sinay Yu.G.: Estimation of the parameters of a complex

stationary Gaussian Markov process, Dokl. Akad. Nauk. SSSR 146(1962), 747–750.4. Balescu, R.: Statistical Dynamics—Matter out of Equilibrium, Imperial College Press, 1997.5. Barros-Neto, J.: An Introduction to the Theory of Distributions, Marcel Dekker, Inc., New

York. 1973.

6. Bell, D. R.: The Malliavin Calculus, Dover Publications, Inc., Mineola, New York. 2006.7. Chen, Y.: Comment on “On the IR-ratio rule of power spectra”[ J. Math. Phys. 50, 063301

(2009)], J. Math. Phys. 51(2010), no.4, 044101.8. Chen, Y. and Liu, Y.: On the eigenfunctions of the complex Ornstein-Uhlenbeck operators,

Kyoto J. Math., 54(2014), no.3, 577–596.9. Ito, K.: Complex multiple Wiener integral, Japan J. Math. 22(1953), 63–86. Reprinted in:

Kiyosi Ito selected papers, Edited by Daniel W. Stroock, S.R.S. Varadhan, Springer-Verlag,

1987.10. Jones, D. S.: The Theory of Generalised Functions, 2nd ed, Cambridge University Press.

1982.11. Kuo, H. H.: Introduction to Stochastic Integration, Springer. 2006.

12. Lunardi, A.: On the Ornstein-Uhlenbeck operator in L2 spaces with respect to invariantmeasures, Trans. Amer. Math. Soc. 349(1997), no.1, 155–169.

13. Malliavin, P.: Stochasitc Analysis, Springer.1997.14. Nualart, D.: The Malliavin Calculus and Related Topics, Springer-Verlag. 2006.

15. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. AMS 44. Springer. 1983.

16. Qian, M.-P. and Xie, J.-S.: On the IR-ratio rule of power spectra, J. Math. Phys. 50(2009),no.6, 063301.

17. Reed, M. and Simon, B.: Methods of Modern Mathematical Physics, New York: AcademicPress, Vol. 1. 1975

18. Riesz, F. and Nagy, B. S.: Functional Analysis, Dover Publications, Inc., New York. 1955.19. Rudin, W.: Functional Analysis, 2nd Ed., McGraw-Hill Companies, Inc. 1991.

20. Shigekawa, I.: Stochasitc Analysis, Translations of Mathematical Monographs, Vol.224. 1998.21. Stein, E. M. and Shakarchi, R.: Complex Analysis, Princeton University Press, Princeton

and Oxford.2003.

22. Stroock, D. W.: The Malliavin calculus, a functional analytic approach, J. Funct. Anal.,44(1981), no.2, 212–257.

23. Stroock, D. W.: Some applications of stochastic calculus to partial differential equations. In:Ecole d’Ete de Probabilites de Saint-Flour XI-1981. Lecture Notes in Math., 976, 268–380,

Springer-Verlag. 1983.

372 YONG CHEN

Y. CHEN: College of Mathematics and Information Science, Jiangxi Normal Uni-

versity, Nanchang, 330022, Jiangxi, P.R.China.E-mail address: [email protected]; [email protected]

AR(1) SEQUENCE WITH RANDOM COEFFICIENTS:

REGENERATIVE PROPERTIES AND ITS APPLICATION

KRISHNA B. ATHREYA, KOUSHIK SAHA, AND RADHENDUSHKA SRIVASTAVA*

Abstract. Let Xnn≥0 be a sequence of real valued random variables such

that Xn = ρnXn−1 + ϵn, n = 1, 2, . . ., where (ρn, ϵn)n≥1 are i.i.d. andindependent of initial value (possibly random) X0. In this paper it is shownthat, under some natural conditions on the distribution of (ρ1, ϵ1), the se-

quence Xnn≥0 is regenerative in the sense that it could be broken up intoi.i.d. components. Further, when ρ1 and ϵ1 are independent, we construct anon-parametric strongly consistent estimator of the characteristic functionsof ρ1 and ϵ1.

1. Introduction

Let Xnn≥0 be a sequence of real valued random variables satisfying the sto-chastic recurrence equation

Xn = ρnXn−1 + ϵn, n = 1, 2, . . . , (1.1)

where (ρn, ϵn)n≥1 are i.i.d. R2-valued random vectors and independent of theinitial random variable X0. If E(|X0|) < ∞ and E(ϵn) = 0, for each n ≥ 1 andthen E(Xn|X0, . . . , Xn−1) = E(ρn)Xn−1. For this reason the sequence Xn sat-isfying (1.1) is often referred to in the time series literature as Random CoefficientAuto Regressive sequence of order one (RCAR(1)) (see [1, 6, 7, 9]). [5] studied aparametric model for (ρ1, ϵ1) under the assumption that ρ1 and ϵ1 are indepen-dent and provided a consistent estimator of the model parameters. In the currentpaper, we find conditions on the distribution function of (ρ1, ϵ1) to ensure thatXn is a Harris recurrent Markov chain and hence regenerative, i.e., it can bebroken up into i.i.d. excursions. We exploit the regenerative property of Xn toconstruct a non-parametric consistent estimator of the characteristic functions ofρ1 and ϵ1 under the independence assumption of ρ1 and ϵ1.

A sequence Xnn≥0 is said to be delayed regenerative if there exists a sequenceTjj≥1 of positive integer valued random variables such that P(0 < Tj+1 − Tj <∞) = 1 for all j ≥ 1 and the random cycles ηj ≡ (Xi : Tj ≤ i < Tj+1, Tj+1−Tj)for j = 1, 2, . . . are i.i.d. and independent of η0 ≡ (Xi : 0 ≤ i < T1, T1). If

Received 2017-9-13; Communicated by Hui-Hsiung Kuo.2010 Mathematics Subject Classification. Primary 60J10; Secondary 62M10.Key words and phrases. Auto-regressive sequence, transition function, Harris recurrence,

regeneration.* This research of Radhendushka Srivastava is partially supported by INSPIRE research grant

of DST, Govt. of India and a seed grant from IIT Bombay.

373



374 K. B. ATHREYA, K. SAHA, AND R. SRIVASTAVA

ηjj≥0 are i.i.d. then Xn is called non-delayed regenerative sequence. If, inaddition, E(T2−T1) <∞ then Xn is called regenerative and positive recurrent.

If Xn is a Markov chain with a general state space (S,S), that is Harris irre-ducible and recurrent (see Definition 3.1) then it can be shown that Xn is regen-erative ([4]). Further if Xn admits a stationary probability measure (necessarilyunique because of irreducibility), then Xn is positive recurrent regenerative aswell.

In Sections 2 and 3, under some condition on the distribution of (ρ1, ϵ1) weshow that the sequence Xn satisfying (1.1) is positive recurrent and regener-ative by establishing that Xn admits a stationary distribution and is Harrisirreducible, respectively. In Section 4, we show that the distribution of (ρ1, ϵ1)can be determined by transition probability function of Xn. We subsequentlyprovide a consistent estimator of transition probability function of Xn by usingthe regenerative property. Finally, if ρ1 and ϵ1 are independent then we provide anon-parametric consistent estimator of characteristic function of ρ1 and ϵ1, basedon Xnn≥0.

2. Limit Distribution of Xn

We begin with existence of the limiting distribution of Xn in (1.1).

Theorem 2.1. Let −∞ ≤ E(log |ρ1|) < 0 and E(log |ϵ1|)+ < ∞. Then Xn in(1.1) converges in distribution to X∞ as n→ ∞ where

X∞ ≡ ϵ1 + ρ1ϵ2 + ρ1ρ2ϵ3 + . . .+ ρ1 . . . ρnϵn+1 + . . . . (2.1)

The infinite series on the right hand side of (2.1) is absolutely convergent withprobability 1.

The above result can be deduced from [6]. A proof of Theorem 2.1 is given inthe appendix. Theorem 2.1 does not indicate nature of limiting distribution ofXn. We show that the distribution of X∞ is non-atomic when the distribution of(ρ1, ϵ1) is non-degenerate.

Theorem 2.2. Let −∞ ≤ E(log |ρ1|) < 0, E(log |ϵ1|)+ < ∞, P(ρ1 = 0) = 0 and(ρ1, ϵ1) has a non-degenerate distribution. Then X∞ has a non atomic distribution,i.e., P(X∞ = a) = 0 for all a ∈ R.

Proof. Since (ρ1, ϵ1) has a nondegenerate distribution, the random variable X∞ asin (2.1) does not have a degenerate distribution and hence supP(X∞ = a) : a ∈R ≡ p < 1. Let a0 be such that P(X∞ = a0) = p. Then by Doob’s martingaleconvergence theorem (see page 211 of [3]), we have

E(I(X∞ = a0)|Fn) → E(I(X∞ = a0)|F∞) w. p. 1 (2.2)

where, Fn ≡ σ(ρi, ϵi) : i = 1, 2, . . . , n,X0, the σ-algebra generated by (ρi, ϵi) fori = 1, . . . , n and X0, and F∞ ≡ σ(ρi, ϵi) : i ∈ N, X0. Since X∞ is measurable

REGENERATIVE PROPERTY OF RCAR(1) 375

with respect to F∞, E(I(X∞ = a0)|F∞) = I(X∞ = a0). Next,

E(I(X∞ = a0)|Fn)

= P(ϵ1 + ρ1ϵ2 + · · ·+ ρ1 · · · ρn−1ϵn + ρ1 · · · ρn(ϵn+1 + ρn+1ϵn+2 + · · · ) = a0|Fn)

= P(Yn =

a0 − ϵ1 − ρ1ϵ2 − · · · − ρ1ρ2 · · · ρn−1ϵnρ1ρ2 · · · ρn

|Fn

)where Yn = ϵn+1 + ρn+1ϵn+2 + ρn+1ρn+2ϵn+3 + · · · and last equality holds sinceP(ρ1 = 0) = 0, |ρ1 · · · ρn| = 0 ∀ n ≥ 1. But Yn and X∞ have the same distribution,

and Yn is independent of Fn and a0−ϵ1−ρ1ϵ2−···−ρ1ρ2···ρn−1ϵnρ1ρ2···ρn

is Fn measurable. So

E(I(X∞ = a0)|Fn) ≤ p < 1 for all n ≥ 1.

From (2.2), it follows that I(X∞ = a0) ≤ p < 1 with probability 1. Since I(X∞ =a0) is a 0, 1 valued random variable, I(X∞ = a0) = 0 with probability 1 andhence P(X∞ = a0) = 0. Hence, X∞ has a non atomic distribution.

A natural question is under what additional conditions on the distribution of(ρ1, ϵ1), the sequence Xn is regenerative. When a Markov sequence is Harrisrecurrent and σ-algebra is countably generated then it can be established thatthe sequence exhibits regenerative property (see [4]). We now explore the Harrisrecurrence property of Xn.

3. Harris Recurrence of Xn

Definition 3.1. A Markov chain Xnn≥0 is called Harris or ϕ-recurrent if thereexists a σ-finite measure ϕ on the state space (S,S) such that

ϕ(A) > 0 =⇒ P(τA <∞|X0 = x) = 1 ∀x ∈ S, (3.1)

where τA = minn : n ≥ 1, Xn ∈ A.

Note that any irreducible and recurrent Markov chain with a countable statespace is Harris recurrent as one can take ϕ to be the δ measure at some i0 ∈ S. Adefinition related to Definition 1 is given by [4].

Definition 3.2. A Markov chain Xn is called (A, ϵ, ϕ, n0) recurrent if thereexists a set A ∈ S, a probability measure ϕ on S, a real number ϵ > 0, and aninteger n0 > 0 such that

P(τA <∞|X0 = x) ≡ Px(τA <∞) = Px(Xn ∈ A for some n ≥ 1) = 1 ∀ x ∈ S(3.2)

and

P(Xn0 ∈ E|X0 = x) ≡ Px(Xn0 ∈ E) = P(n0)(x,E) ≥ ϵϕ(E) ∀ x ∈ A , ∀ E ⊂ S.(3.3)

It can be shown by using the C-set lemma of Doob (see [8]) that when S iscountably generated, then Definition 3.1 implies Definition 3.2. That Definition3.2 implies Definition 3.1 is not difficult to prove.

The following theorem provides a sufficient condition for Xn in (1.1) to be aHarris recurrent Markov chain.


Theorem 3.3. Let −∞ ≤ E(log |ρ1|) < 0, E(log |ϵ1|)+ < ∞, P(ρ1 = 0) = 0 and−∞ < c < d <∞ be such that P(c ≤ X∞ ≤ d) > 0. Then for all x ∈ R,

P(Xn ∈ [c, d] for some n ≥ 1|X0 = x) = 1. (3.4)

In addition, let there exists a finite measure ϕ on R such that ϕ([c, d]) > 0 and0 < α < 1 such that

infc≤x≤d

P(ρ1x+ ϵ1 ∈ ·) ≥ αϕ(·). (3.5)

Then the Markov chain Xnn≥0 as described in (1.1) is Harris recurrent andhence regenerative.

Note that since Xn converges in distribution to X∞ which is a proper realvalued random variable, Xnn≥0 is positive recurrent as well. Thus under thehypothesis of Theorem 3.3, Xnn≥0 is regenerative and positive recurrent. Theproof of Theorem 3.3 is based on the following results.

Lemma 3.4. Let −∞ ≤ E(log |ρ1|) < 0, E(log |ϵ1|)+ < ∞, P(ρ1 = 0) = 0 and−∞ < c < d < ∞ be such that P(c ≤ X∞ ≤ d) > 0. Then there exist θ > 0 andfor all x ∈ R, an integer nx ≥ 1 such that

P(Xn ∈ [c, d]|X0 = x) ≥ θ for all n ≥ nx. (3.6)

Proof. Iterating (1.1) yields,

Xn = ρnρn−1 · · · ρ1X0 + ρnρn−1 · · · ρ2ϵ1 + · · ·+ ρnϵn−1 + ϵn ≡ ZnX0 + Yn, say.

So, if X0 = x w. p. 1, then

Px(Xn ∈ [c, d]) = P(Yn + Znx ∈ [c, d])

≥ P(Yn ∈ [c+ η, d− η], |Znx| < η)

≥ P(Yn ∈ [c+ η, d− η])− P(|Znx| ≥ η),

where η > 0 such that c+ η < d− η. Now, define

Y ′n ≡ ϵ1 + ρ1ϵ2 + ρ1ρ2ϵ3 + · · ·+ ρ1 . . . ρn−1ϵn. (3.7)

Note that the distribution of Yn and Y ′n are same and from Theorem 2.1, Y ′

n → X∞with probability 1. Thus, we have

Px(Xn ∈ [c, d])

≥ P(Y ′n ∈ [c+ η, d− η])− P(|Znx| ≥ η)

≥ P(Y ′n ∈ [c+ η, d− η], |Y ′

n −X∞| ≤ η′)− P(|Znx| ≥ η)

≥ P(X∞ ∈ [c+ η + η′, d− η − η

′])− P(|Y ′

n −X∞| ≥ η′)− P(|Znx| ≥ η),

where η′> 0 such that c+ η + η

′< d− η − η

′.

Now choose n1 large such that P(|Y ′n −X∞| ≥ η

′) ≤ δ

2 and n2 large such that

P(|Zn2x| ≥ η) ≤ δ2 . Note that choice of n2 depends on x. Let nx = max(n1, n2).

Then for all n ≥ nx,

Px(Xnx ∈ [c, d]) ≥ P(X∞ ∈ [c+ η + η′, d− η − η

′])− δ.


Since X∞ has a continuous distribution by Theorem 2.2 and P(c ≤ X∞ ≤ d) > 0,

first choose η and η′and then δ small enough such that

θ ≡ P(X∞ ∈ [c+ η + η′, d− η − η

′])− δ > 0.

Thus (3.6) is established.

Lemma 3.5. Let Xn be a time homogeneous Markov chain with state space(S,S) and transition function P (·, ·). Let there exists A ∈ S and 0 < θ ≤ 1 suchthat for all x ∈ S, there exists an integer nx ≥ 1 such that

P(Xnx ∈ A|X0 = x) ≥ θ. (3.8)

Then for all x ∈ S,

P(τA <∞|X0 = x) = 1 (3.9)

where τA = minn : n ≥ 1, Xn ∈ A).

Proof. Fix x ∈ S. Let B0 ≡ Xnx /∈ A and τ0 = nx. Then B0 ≡ Xτ0 /∈ A. Letus define

B1 ≡ Xτ0 /∈ A, Xτ0+nXτ0/∈ A

τ1 = τ0 + nXτ0

B2 ≡ Xτ0 /∈ A, Xτ1 /∈ A, Xτ1+nXτ1/∈ A

τ2 = τ1 + nXτ1,

and so on. Note B1 = Xτ0 /∈ A,Xτ1 /∈ A, B2 = Xτ0 /∈ A,Xτ1 /∈ A,Xτ2 /∈ Aand for any integer k ≥ 3,

Bk ≡ Xτ0 /∈ A,Xτ1 /∈ A, . . . ,Xτk /∈ A,

with τk = τk−1 + nXτk−1. By hypothesis (3.8), P((B0) ≤ (1 − θ). By the strong

Markov property of Xn, P(B1) ≤ (1− θ)2 and P(Bk) ≤ (1− θ)k+1 for all integerk ≥ 3. This implies

∑∞k=0 P(Bk) < ∞ since θ > 0. So

∑∞k=0 IBk

(·) < ∞ withprobability 1. This implies that with probability 1, IBk

= 0 for all large k > 1.That is, for all x ∈ S, Px(Xτk ∈ A for some k < ∞) = 1. Hence, for all x ∈ S,Px(τA <∞) = 1.

Proof of Theorem 3.3. In view of Definition 3.2, it is enough to prove (3.4) to showXn is Harris recurrent. The proof of (3.4) follows from Lemma 3.4 and 3.5. Nowfrom Lemma 2.2.5 of [2], it follows that Xn is regenerative.

Theorem 3.3 provides sufficient conditions on (ρ1, ϵ1) so that the sequence Xn

becomes Harris recurrent and hence regenerative. These sufficient conditions arefairly general and hold for large class of distribution of (ρ1, ϵ1). Here are someexamples where (3.4) and (3.5) hold.

Example 3.6. ϵ1 is a standard normal, N(0, 1) random variable, ρ1 has boundedsupport with E log |ρ1| < 0 and ϵ1, ρ1 are independent.

Example 3.7. ϵ1 is a Uniform (−1, 1) random variable, ρ1 has bounded supportwith E log |ρ1| < 0 and ϵ1, ρ1 are independent.


In both the cases hypothesis of Theorem 2.1 hold and X∞ is of the form (ϵ1 +

ρ1X∞) where X∞ has the same distribution as X∞ and independent of X∞. Onecan show in both above cases that for some c < 0 < d, |c| and d sufficiently small,conditions (3.4) and (3.5) hold.

In Theorem 3.3, growth sequence ρn has no mass at zero and the regenerationproperty of Xn is established by showing Harris recurrence of the sequence. WhenP(ρ1 = 0) > 0, then the regenerative property of Xn can be shown more easily.

Theorem 3.8. Let Xnn≥0 be a RCAR(1) sequence as in (1.1). If P(ρ1 = 0) ≡α > 0, then Xnn≥0 is a positive recurrent regenerative sequence.

Proof. Let τ0 = 0 and τj+1 = minn : n ≥ τj + 1, ρn = 0 for j ≥ 0. We need toshow that

P(τj+1 − τj = kj , Xτj+l ∈ Al,j , 0 ≤ l < kj , 1 ≤ j ≤ r)

=r∏

j=1

P(τ2 − τ1 = kj , Xτ1+l ∈ Al,j , 0 ≤ l < kj) (3.10)

for all k1, k2, . . . , kr ∈ N and Al,j ∈ B(R), 0 ≤ l < kj , j = 1, 2, . . . , r, r = 1, 2, . . ..Since (ρn, ϵn)n≥1 are i.i.d. and P(ρ1 = 0) = α > 0, it follows that τj+1 −

τj , j ≥ 0 are i.i.d. with jump distribution

P(τj+1 − τj = k) = (1− α)k−1α, for k = 1, 2, . . . ,

that is, geometric with “success” parameter α. Next, since (ρn, ϵn)n≥1 are i.i.d.(3.10) follows. Further since E(τ2 − τ1) < ∞, the sequence Xn is positiverecurrent regenerative. Remark 3.9. When P(ρ1 = 0) = α > 0 and the joint distribution of (ρ1, ϵ1) isdiscrete, then the limiting distribution π of X∞ is a discrete probability distribu-tion, that is, there exists a countable set A0 in R2 such that π(A0) = 1. This isin contrast to Theorem 2.2 which provides a sufficient condition for X∞ to have anon atomic distribution.

4. Estimation of Transition Function and Characteristic Functionsof ρ1 and ϵ1

The transition function P(x,A) of the Markov chain Xnn≥0, defined by (1.1),is precisely equal to P(ρ1x + ϵ1 ∈ A). The following result determines the jointdistribution of (ρ1, ϵ1) in terms of the transition function, P(·, ·).Theorem 4.1. If the distribution of ρ1x+ϵ1 is known for all x of the form t1

t2where

t2 = 0 and (t1, t2) is dense in R2 then the distribution of (ρ1, ϵ1) is determined.

Proof. For any (t1, t2) ∈ R2, the characteristic function of (ρ1, ϵ1) is

ψ(ρ1,ϵ1)(t1, t2) = E(ei(t1ρ1+t2ϵ1)) = E(eit2(ρ1t1t2

+ϵ1)) = ϕ t1t2

(t2)

where ϕx(t) = E(eit(ρ1x+ϵ1)) for all x, t ∈ R. If ϕx(·) is known for all x of theform t1

t2where (t1, t2) is dense in R2, then ψ(ρ1,ϵ1)(t1, t2) is determined for all such

(t1, t2) and hence by continuty for all (t1, t2) ∈ R2. Hence the distribution of(ρ1, ϵ1) is determined completely.


Theorem 4.1 implies that if the transition function P(x,A) of Xnn≥0 canbe determined from observing the sequence sequence Xn, then the distributionof (ρ1, ϵ1) can also be determined. We now estimate the transition probabilityfunction P(x, (−∞, y]), for x, y ∈ R2 from the data Xini=0. In the followingtheorems, we show that the estimator Fn,h(x, y), given in (4.1) below, is a stronglyconsistent estimator for P(X1 ≤ y|X0 = x).

Theorem 4.2. Let Xn satisfies the hypothesis of Theorem 3.3. For n ≥ 1,h > 0, x, y ∈ R, let

Fn,h(x, y) =

1nh

∑n−1i=0 I(x≤Xi≤x+h,Xi+1≤y)1

nh

∑ni=0 I(x≤Xi≤x+h)

if I(x ≤ Xi ≤ x+ h) = 0 for some i

0 otherwise,

(4.1)where I(A) denotes the indicator function of the event A.

(a) Then with probability 1, for each x, y ∈ R

limn→∞

Fn,h(x, y) ≡ ψ(x, y, h) =

∫ x+h

xG(u, y)P(X∞ ∈ du)

P(X∞ ∈ (x, x+ h]), (4.2)

where G(u, y) = P(x, (−∞, y]) = P(X1 ≤ y|X0 = x).(b) In addition, let G(x, y) and the random variable X∞ satisfy

limh→0

∫ x+h

xG(u, y)P(X∞ ∈ du)

P(x < X∞ ≤ x+ h)= G(x, y), for x, y ∈ R2. (4.3)

Then for x, y ∈ R

limh→0

limn→0

Fn,h(x, y) = P(X1 ≤ y|X0 = x), with probability 1. (4.4)

Proof. Since Xii≥0 is regenerative and positive recurrent, the vector sequence(Xi, Xi+1)i≥0 is also regenerative and positive recurrent Markov chain. The

numerator in (4.1) converges to∫ x+h

xG(u, y)P(X∞ ∈ du) with probability 1 by

using Theorem 9.2.10 of [3]. Similarly denominator converges to P(X∞ ∈ (x, x+h])with probability 1. This completes the proof of part (a).

The proof of part (b) follows from (4.2) and (4.3).

Remark 4.3. A sufficient condition for (4.3) to hold is that the distribution of X∞is absolutely continuous with strictly positive and continuous density function andthe function G(x, y) is continuous in x for fixed y.

The following result is similar to that of Theorem 4.2.

Theorem 4.4. Fix x, t, h ∈ R. Let

ϕn,h,x(t) =

1

nh

∑n−1j=0 eitXj+1 I(x<Xj≤x+h)

1nh

∑n−1j=0 I(x<Xj≤x+h)

if I(x ≤ Xi ≤ x+ h) = 0 for some i,

0 otherwise.

Then

limh→0

limn→∞

ϕn,h,x(t) = E(eit(ρ1x+ϵ1)) with probability 1,


provided

limh→0

1h

∫ x+h

xE(eit(ρ1x+ϵ1))P(X∞ ∈ du)

1h

∫ x+h

xP(X∞ ∈ du)

= E(eit(ρ1x+ϵ1)). (4.5)

Proof. Proof of this theorem is similar to the proof of Theorem 4.2 and henceomitted.

Remark 4.5. A sufficient condition for (4.5) to hold is that the distribution of X∞is absolutely continuous with strictly positive and continuous density function on(−∞,∞) and the function E(eit(ρ1x+ϵ1)) is continuous in x for fixed t.

Let ρ1 and ϵ1 are independent random variables.Then

ϕx(t) ≡ Ex(eitX1) = E(eit(ρ1x+ϵ1)) = ψρ(tx)ψϵ(t),

where ψρ(t) = E(eitρ) and ψϵ(t) = E(eitϵ). Also, note that

ψϵ(t) = ϕ0(t) and ψρ(tx) =ϕx(t)

ϕ0(t), when ψϵ(t) = 0.

This yields the following corollary of Theorem 4.4.

Corollary 4.6. Let ρ1 and ϵ1 be independent and conditions of Theorem 4.4 holds.Then

(a) limh→0 limn→∞ ϕn,h,0(t) = ψϵ(t) for all t ∈ R with probability 1.(b) Let ψϵ(t) = 0 for all t ∈ R, then for all x = 0

limh→0

limn→∞

ϕn,h,x(t/x)

ϕn,h,0(t/x)= ψρ(t) for all t ∈ R with probability 1.

5. Appendix

Proof of Theorem 2.1: Choose ϵ > 0 such that E(log |ρ1|) + ϵ < 0. Now, by thestrong law of large number,

E(log |ρ1|) < 0 ⇒ 1

n

n∑i=1

log |ρi| ≤ E(log |ρ1|) + ϵ,

for sufficiently large n, with probability 1. Hence

|ρ1ρ2 . . . ρn| ≤ e−nλ, (5.1)

where 0 < λ ≡ −(E(log |ρ1|+ ϵ) <∞, for all large n, with probability 1.Also E(log |ϵ1|)+ < ∞ implies that for any µ > 0,

∑∞n=1 P(log |ϵ1| > nµ) < ∞

and hence∑

n P(log |ϵn| > nµ) < ∞. By Borel Cantelli lemma, |ϵn| ≤ enµ for alln large enough, with probability 1.

Now choose 0 < µ < λ. Then for sufficiently large n, with probability 1,

|ϵn+1ρ1ρ2 . . . ρn| ≤ e−nλe(n+1)µ.

Therefore∑

n |ϵn+1|ρ1ρ2 . . . ρn| <∞ with probability 1. Hence X∞ = ϵ1 + ρ1ϵ2 +ρ1ρ2ϵ3 + . . .+ ρ1 . . . ρnϵn+1 + . . . is well defined.


Observe that

Xn = ρn(ρn−1Xn−2 + ϵn−1) + ϵn

= ρnρn−1 · · · ρ1X0 + ρnρn−1 · · · ρ2ϵ1 + · · ·+ ρnϵn−1 + ϵn

and which has the same distribution as

ϵ1 + ρ1ϵ2 + · · ·+ ρ1ρ2 · · · ρn−1ϵn + ρ1ρ2 · · · ρnX0. (5.2)

Now by using (5.1) and above, we have |ρ1ρ2 · · · ρnX0| converges to zero withprobability 1. Thus, from (5.2), as n→ ∞, we have

Xnd→ X∞,

whered→ stands for convergence in distribution.

Acknowledgment. K. B. Athreya would like to thank the Department of Math-ematics, IIT Bombay and in particular, Prof. Sudhir Ghorpade for offering himvisiting professorship.

References

1. Andel, J.: Autoregressive series with random parameters, Math. Operationsforsch. Statist.7 (1976), no. 5, 735–741.

2. Athreya, K. B. and Atuncar, G. S.: Kernel estimation for real-valued Markov chains, Sankhya

Ser. A 60 (1998), no. 1, 1–17.3. Athreya, K. B. and Lahiri, S. N.: Probaility Theory (TRIM Series 41), Hindustan Book

agency, New Delhi, 2006.4. Athreya, K. B. and Ney, P.: A new approach to the limit theory of recurrent Markov chains,

Trans. Amer. Math. Soc. 245 (1978), 493–501.5. Aue, A., Horvath, L., and Steinebach, J.: Estimation in random coefficient autoregressive

models, J. Time Ser. Anal. 27 (2006), no. 1, 61–76.6. Brandt, A.: The stochastic equation Yn+1 = AnYn + Bn with stationary coefficients, Adv.

in Appl. Probab. 18 (1986), no. 1, 211–220.7. Nicholls, D. F. and Quinn, B. G.: The estimation of random coefficient autoregressive models

I, J. Time Ser. Anal. 1 (1980), no. 1, 37–46.8. Orey, S.: Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities, Van

Nostrand Reinhold Co., London-New York-Toronto, Ont., 1971.9. Robinson, P. M.: Statistical inference for a random coefficient autoregressive model, Scand.

J. Statist. 5 (1978), no. 3, 163–168.

Krishna B. Athreya: Departments of Mathematics and Statistics, Iowa State Uni-versity, Iowa 50011, USA and Distinguished Visiting Professor, Department of Math-

ematics, IIT Bombay, Mumbai 400076, IndiaE-mail address: [email protected]

Koushik Saha: Department of Mathematics, Indian Institute of Technology Bom-bay, Mumbai 400076, India


Radhendushka Srivastava: Department of Mathematics, Indian Institute of Tech-nology Bombay, Mumbai 400076, India


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