re gresi on based montecarlo methods

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arXiv:math/0

508491v1

[math.P

R]25Aug2005

The Annals of Applied Probability

2005, Vol. 15, No. 3, 21722202DOI: 10.1214/105051605000000412

c Institute of Mathematical Statistics, 2005

A REGRESSION-BASED MONTE CARLO METHOD TO SOLVEBACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS1

By Emmanuel Gobet, Jean-Philippe Lemor and Xavier Warin

Centre de Mathematiques Appliquees, Electricite de France and Electricitede France

We are concerned with the numerical resolution of backwardstochastic differential equations. We propose a new numerical schemebased on iterative regressions on function bases, which coefficients areevaluated using Monte Carlo simulations. A full convergence analy-

sis is derived. Numerical experiments about finance are included, inparticular, concerning option pricing with differential interest rates.

1. Introduction. In this paper we are interested in numerically approxi-mating the solution of a decoupled forwardbackward stochastic differentialequation (FBSDE)

St = S0 +

t0

b(s, Ss) ds +

t0

(s, Ss) dWs,(1)

Yt = (S) +

Tt

f(s, Ss, Ys, Zs) dsT

tZs dWs.(2)

In this representation,S

= (St : 0 t T) is the d-dimensional forward com-ponent and Y = (Yt : 0 t T) the one-dimensional backward one (the ex-tension of our results to multidimensional backward equations is straight-forward). Here, W is a q-dimensional Brownian motion defined on a filteredprobability space (,F,P, (Ft)0tT), where (Ft)t is the augmented naturalfiltration of W. The driver f(, , , ) and the terminal condition () are,respectively, a deterministic function and a deterministic functional of theprocess S. The assumptions (H1)(H3) below ensure the existence and theuniqueness of a solution (S,Y,Z) to such equation (1)(2).

Received June 2004; revised January 2005.1Supported by Association Nationale de la Recherche Technique, Ecole Polytechnique

and Electricite de France.AMS 2000 subject classifications. 60H10, 60H10, 65C30.Key words and phrases. Backward stochastic differential equations, regression on func-

tion bases, Monte Carlo methods.

This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in The Annals of Applied Probability,2005, Vol. 15, No. 3, 21722202. This reprint differs from the original inpagination and typographic detail.

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2 E. GOBET, J.-P. LEMOR AND X. WARIN

Applications of BSDEs. Such equations, first studied by Pardoux andPeng [26] in a general form, are important tools in mathematical finance.We mention some applications and refer the reader to [10, 12] for numerousreferences. In a complete market, for the usual valuation of a contingentclaim with payoff (S), Y is the value of the replicating portfolio and Z isrelated to the hedging strategy. In that case, the driver f is linear w.r.t. Yand Z. Some market imperfections can also be incorporated, such as higherinterest rate for borrowing [4]: then, the driver is only Lipschitz continuousw.r.t. Y and Z. Related numerical experiments are developed in Section 6.In incomplete markets, the FollmerSchweizer strategy [14] is given by thesolution of a BSDE. When trading constraints on some assets are imposed,the super-replication price [13] is obtained as the limit of nonlinear BSDEs.Connections with recursive utilities of Duffie and Epstein [11] are also avail-able. Peng has introduced the notion of g-expectation (here g is the driver)as a nonlinear pricing rule [28]. Recently he has shown [27] the deep connec-tion between BSDEs and dynamic risk measures, proving that any dynamicrisk measure (Et)0tT (satisfying some axiomatic conditions) is necessarilyassociated to a BSDE (Yt)0tT (the converse being known for years). Theleast we can say is that BSDEs are now inevitable tools in mathematicalfinance. Another indirect application may concern variance reduction tech-niques for the Monte Carlo computations of expectations, say E() taking

f 0. Indeed, T0 Zs dWs is the so-called martingale control variate (see [24],for instance). Finally, for applications to semi-linear PDEs, we refer to [25],

among others.The mathematical analysis of BSDE is now well understood (see [23] for

recent references) and its numerical resolution has made recent progresses.However, even if several numerical methods have been proposed, they sufferof a high complexity in terms of computational time or are very costly interms of computer memory. Thus, their uses in practice on real problems aredifficult. Hence, it is still topical to devise more efficient algorithms. Thisarticle contributes in this direction by developing a simple approach, basedon Monte Carlo regression on function bases. It is in the vein of the generalregression approach of Bouchard and Touzi [6], but here it is actually muchsimpler because only one set of paths is used to evaluate all the regressionoperators. Consequently, the numerical implementation is easier and more

efficient. In addition, we provide a full mathematical analysis of the influenceof the parameters of the method.

Numerical methods for BSDEs. In the past decade, there have been sev-eral attempts to provide approximation schemes for BSDEs. First, Ma, Prot-ter and Yong [22] propose the four step scheme to solve general FBSDEs,which requires the numerical resolution of a quasilinear parabolic PDE. In


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MONTE CARLO METHOD FOR BSDE 3

[2], Bally presents a time discretization scheme based on a Poisson net:this trick avoids him using the unknown regularity of Z and enables himto derive a rate of convergence w.r.t. the intensity of the Poisson process.However, extra computations of very high-dimensional integrals are neededand this is not handled in [2]. In a recent work [29], Zhang proves some L2-regularity on Z, which allows the use of a regular deterministic time mesh.Under an assumption of constructible functionals for (which essentiallymeans that the system can be made Markovian, by adding d extra statevariables), its approximation scheme is less consuming in terms of high-dimensional integrals. If for each of the d + d state variables, one uses Mpoints to compute the integrals, the complexity is about Md+d

per time

step, for a global error of order M1 say (actually, an analysis of the global

accuracy is not provided in [29]). This approach is somewhat related to thequantization method of Bally and Pages [3], which is an optimal space dis-cretization of the underlying dynamic programming equation (see also theformer work by Chevance [8], where the driver does not depend on Z). Weshould also mention the works by Ma, Protter, San Martin and Soledad [21]and Briand, Delyon and Memin [7], where the Brownian motion is replacedby a scaled random walk. Weak convergence results are given, without ratesof approximation. The complexity becomes very large in multidimensionalproblems, like for finite differences schemes for PDEs. Recently, in the caseof path-independent terminal conditions (S) = (ST), Bouchard and Touzi[6] propose a Monte Carlo approach which may be more suitable for high-dimensional problems. They follow the approach by Zhang [29] by approx-

imating (1)(2) by a discrete time FBSDE with N time steps [see (5)(6)below], with an L2-error of order N1/2. Instead of computing the condi-

tional expectations which appear at each discretization time by discretizingthe space of each state variable, the authors use a general regression opera-tor, which can be derived, for instance, from kernel estimators or from theMalliavin calculus integration by parts formulas. The regression operator ata discretization time is assumed to be built independently of the underlyingprocess, and independently of the regression operators at the other times.For the Malliavin calculus approach, for example, this means that one needsto simulate at each discrete time, M copies of the approximation of (1),which is very costly. The algorithm that we propose in this paper requiresonly one set of paths to approximate all the regression operators at each dis-cretization time at once. Since the regression operators are now correlated,the mathematical analysis is much more involved.

The regression operator we use in the sequel results from the L2-projectionon a finite basis of functions, which leads in practice to solve a standardleast squares problem. This approach is not new in numerical methods forfinancial engineering, since it has been developed by Longstaff and Schwartz[20] for the pricing of Bermuda options. See also [5] for the option pricingusing simulations under the objective probability.


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Organization of the paper. In Section 2 we set the framework of ourstudy, define some notation used throughout the paper and describe ouralgorithm based on the approximation of conditional expectations by a pro-jection on a finite basis of functions. We also provide some remarks relatedto models in finance.

The next three sections are devoted to analyzing the influence of theparameters of this scheme on the evaluation of Y and Z. Note that approx-imation results on Z were not previously considered in [6]. In Section 3 weprovide an estimation of the time discretization error: this essentially fol-lows from the results by Zhang [29]. Then, the impact of the function basesand the number of simulated paths is separately discussed in Section 4 andin Section 5, which is the major contribution of our work. Since this leastsquares approach is also popular to price Bermuda options [20], it is crucialto accurately estimate the propagation of errors in this type of numericalmethod, that is, to ensure that it is not explosive when the exercise fre-quency shrinks to 0. L2-estimates and a central limit theorem (see also [9]for Bermuda options) are proved.

In Section 6 explicit choices of function bases are given, together with nu-merical examples relative to the pricing of vanilla options and Asian optionswith differential interest rates.

2. Assumptions, notation and the numerical scheme.

2.1. Standing assumptions. Throughout the paper we assume that thefollowing hypotheses are fulfilled:

(H1) The functions (t, x) b(t, x) and (t, x) (t, x) are uniformly Lips-chitz continuous w.r.t. (t, x) [0, T]Rd.

(H2) The driver f satisfies the following continuity estimate:

|f(t2, x2, y2, z2) f(t1, x1, y1, z1)|Cf(|t2 t1|1/2 + |x2 x1|+ |y2 y1|+ |z2 z1|)

for any (t1, x1, y1, z1), (t2, x2, y2, z2) [0, T]Rd RRq.(H3) The terminal condition satisfies the functional Lipschitz condition,

that is, for any continuous functions s1 and s2, one has

|(s1)(s2)| C supt[0,T]

|s1t s2t |.

These assumptions (H1)(H3) are sufficient to ensure the existence anduniqueness of a triplet (S,Y,Z) solution to (1)(2) (see [23] and referencestherein). In addition, the assumption (H3) allows a large class of terminalconditions (see examples in Section 2.4).


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To approximate the forward component (1), we use a standard Eulerscheme with time step h (say smaller than 1), associated to equidistantdiscretization times (tk = kh = kT/N)0kN. This approximation is definedby SN0 = S0 and

SNtk+1 = SNtk

+ b(tk, SNtk

)h + (tk, SNtk

)(Wtk+1 Wtk).(3)The terminal condition (S) is approximated by N(PNtN), where

N is

a deterministic function and (PNtk )0kN is a Markov chain, whose firstcomponents are given by those of (SNtk )0kN. In other words, we even-tually add extra state variables to make Markovian the implicit dynam-ics of the terminal condition. We also assume that PNtk is Ftk -measurable

and thatE

[

N

(P

N

tN)]

2


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denoted 0,k, 1,k, . . . , q,k (viewed as column vectors). We assume thatE|pl,k|2


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Associated to these estimates, we define (random) truncation functionsNl,k(x) =

Nl,k(P

Ntk

)(x/Nl,k(PNtk

)) and N,ml,k (x) = Nl,k(P

N,mtk

)(x/Nl,k(PN,mtk

)),

where :R R is a C2b -function, such that (x) = x for |x| 3/2, || 2and || 1.

In the next computations, C denotes a generic constant that may changefrom line to line. It is still uniform in the parameters of our scheme.

2.3. The numerical scheme. We are now in a position to define thesimulation-based approximations of the BSDE (1)(2). The statements ofapproximation results and their proofs are postponed to Sections 3, 4 and 5.

Our procedure combines a backward in time evaluation (from time tN = T

to time t0 = 0), a fixed point argument (using i = 1, . . . , I Picard iterations),least squares problems on M simulated paths (using some function bases).

Initialization. The algorithm is initialized with YN,i,I,MtN = N(PNtN) (in-

dependently of i and I). Then, the solution (Ytk , Z1,tk , . . . , Z q,tk) at a given

time tk is represented via some projection coefficients (i,I,Ml,k )0lq by

YN,i,I,Mtk =

N0,k(

i,I,M0,k p0,k),

hZ

N,i,I,Ml,tk

= Nl,k(

hi,I,Ml,k pl,k)

(N0,k and Nl,k are the truncations introduced before). We now detail how the

coefficients are computed using independent realizations ((PN,mtk )0kN)1mM,

((W

m

k )0kN1)1mM.Backward in time iteration at time tk < T. Assume that an approxi-

mation YN,I,I,Mtk+1

:= N0,k+1(I,I,M0,k+1 p0,k+1) is built, and denote YN,I,I,M,mtk+1 =

N,m0,k+1(I,I,M0,k+1 pm0,k+1) its realization along the mth simulation.

For the initialization i = 0 of Picard iterations, set YN,0,I,Mtk = 0 andZ

N,0,I,Mtk = 0, that is,

0,I,Ml,k = 0 (0 l q).

For i = 1, . . . , I , the coefficients i,I,Mk = (i,I,Ml,k )0lq are iteratively ob-tained as the arg min in (0, . . . , q) of the quantity

1

M

Mm=1

Y

N,I,I,M,mtk+1 0 p

m0,k + hf

mk (

i1,I,M

k )q

l=1l p

ml,kW

ml,k2

.(4)

If the above least squares problem has multiple solutions (i.e., the empiricalregression matrix is not invertible, which occurs with small probability whenM becomes large), we may choose, for instance, the (unique) solution ofminimal norm. Actually, this choice is arbitrary and has no incidence on thefurther analysis.


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The convergence parameters of this scheme are the time step h (h 0),the function bases, the number of simulations M (M+). This is fullyanalyzed in the following sections, with three main steps: time discretizationof the BSDE, projections on bases functions in L2(,P), empirical projec-tions using simulated paths. An estimate of the global error directly followsfrom the combination of Theorems 1, 2 and 3. We will also see that it isenough to have I= 3 Picard iterations (see Theorem 3).

The intuition behind the above sequence of least squares problems (4)is actually simple. It aims at mimicking what can be ideally done with aninfinite number of simulations, Picard iterations and bases functions, thatis,

(YNtk , ZNtk ) = arg inf (Y,Z)L2(Ftk )

E(YNtk+1 Y + hf(tk, SNtk , Y , Z )ZWk)2

,

where, as usual, L2(Ftk) stands for the square integrable and Ftk -measurable,possibly multidimensional, random variables. This ideal case is an appoxi-mation of the BSDE (2) which writes

Ytk+1 +

tk+1tk

f(s, Ss, Ys, Zs) ds = Ytk +

tk+1tk

Zs dWs

over the time interval [tk, tk+1]. (YNtk )k will be interpreted as a discrete time

BSDE (see Theorem 1).

2.4. Remarks for models in finance. Here, we give examples of driversf and terminal conditions (S) in the case of option pricing with differentinterest rates [4]: R for borrowing and r for lending with R r. Assumefor simplicity that there is only one underlying risky asset (d = 1) whosedynamics is given by the BlackScholes model with drift and volatility (q= 1): dSt = St( dt + dWt).

Driver: If we set f(t,x,y,z) = {yr + z (y z )(R r)}, where =r

, Yt is the value at time t of the self-financing p ortfolio replicating thepayoff (S) [12]. In the case of equal interest rates R = r, the driver islinear and we obtain the usual risk-neutral valuation rule.

Terminal conditions: A large class of exotic payoffs satisfies the functionalLipschitz condition (H3). Vanilla payoff: (S) = (ST). Set PNtk = SNtk and N(PNtN) = (PNtN).

Under (H3), it gives E|N(PNtN)(S)|2 Ch. Asian payoff: (S) = (ST,

T0 St dt). Set P

Ntk = (S

Ntk , h

k1i=0 S

Nti ) and

N(PNtN) = (PNtN). For usual functions , the L2-error is of order 1/2

w.r.t. h. More accurate approximations of the average of S could beincorporated [18].


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Lookback payoff: (S) = (ST, mint[0,T] St, maxt[0,T] St). Set N(PNtN) =(PNtN) with P

Ntk

= (SNtk , minik SNti , maxik S

Nti ). In general, this in-

duces an L2-error of magnitude

h log(1/h) [29]. The rate

h canbe achieved by considering the exact extrema of the continuous Eulerscheme [1].

Note also that (H4) is satisfied on these payoffs.

We also mention that the price process (St)t is usually positive coordinate-wise, but its Euler scheme [defined in (3)] does not enjoy this feature. Thismay be an undesirable property, which can be avoided by considering theEuler scheme on the log-price. With this modification, the analysis below isunchanged and we refer to [15] for details.

3. Approximation results: step 1. We first consider a time approxima-tion of equations (1) and (2). The forward component is approximated us-ing the Euler scheme (3) and the backward component (2) is evaluated ina backward manner. First, we set YNtN =

N(PNtN). Then, (YNtk

, ZNtk )0kN1are defined by

ZNl,tk =1

hEk(Y

Ntk+1

Wl,k),(5)

YNtk = Ek(YNtk+1

) + hf(tk, SNtk

, YNtk , ZNtk

).(6)

Using, in particular, the inequality

|ZNl,tk

| 1hEk(YNtk+1 )2, it is easy to

see by a recursive argument that YNtk and ZNtk

belong to L2(Ftk). It is alsoequivalent to assert that they minimize the quantity

E(YNtk+1 Y + hf(tk, SNtk , Y , Z )ZWk)2(7)

over L2(Ftk ) random variables (Y, Z). Note that YNtk is well defined in (6),because the mapping Y Ek(YNtk+1 ) + hf(tk, SNtk , Y , Z Ntk ) is a contraction inL2(Ftk), for h small enough. The following result provides an estimate ofthe error induced by this first step.

Theorem 1. Assume (H1)(H3). For h small enough, we have

max0kN

E|Ytk YNtk |2 +N1k=0

tk+1

tk

E|Zt ZNtk |2 dt

C((1 + |S0|2)h +E|(S)N(PNtN)|2).

Proof. From [29], we know that the key point is the L2-regularity ofZ.Here, under (H1)(H3), Z is cadlag (see Remark 2.6.ii in [29]). Thus, The-


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orem 3.1 in [29] states thatN1k=0

E

tk+1tk

|Zt Ztk |2 dtC(1 + |S0|2)h.

With this estimate, the proof of Theorem 1 is standard (see, e.g., the proofof Theorem 5.3 in [29]) and we omit details.

Owing to the Markov chain (PNtk )0kN, the independent increments(Wk)0kN1 and (5)(6), we easily get the following result.

Proposition 1. Assume (H1)(H3). For h small enough, we have

YNtk = yNk (PNtk ), ZNl,tk = zNl,k(PNtk ) for 0 k N and 1 l q,(8)where (yNk ())k and (zNl,k())k,l are measurable functions.

It will be established in Section 6 that they are Lipschitz continuous underthe extra assumption (H4).

4. Approximation results: step 2. Here, the conditional expectationswhich appear in the definitions (5)(6) of YNtk and Z

Nl,tk

(1 l q) are re-placed by a L2(,P) projection on the function bases p0,k and pl,k (1 l q).A numerical difficulty still remains in the approximation ofYNtk in (6), whichis usually obtained as a fixed point. To circumvent this problem, we propose

a solution combining the projection on the function basis and I Picard iter-ations. The integer I is a fixed parameter of our scheme (the analysis belowshows that the value I= 3 is relevant).

Definition 1. We denote by YN,i,Itk the approximation of YNtk

, wherei Picard iterations with projections have been performed at time tk and IPicard iterations with projections at any time after tk. Analogous notationstands for Z

N,i,Il,tk

. We associate to YN,i,Itk and Z

N,i,Il,tk

their respective projec-

tion coefficients i,I0,k and i,Il,k, on the function bases p0,k and pl,k (1 l q).

We now turn to a precise definition of the above quantities. We setYN,i,ItN =

N(PNtN), independently ofi and I. Assume that YN,I,Itk+1

is obtained

and let us define YN,i,Itk , ZN,i,Il,tk

for i = 0, . . . , I . We begin with YN,0,Itk = 0 and

ZN,0,Itk = 0, corresponding to

0,Il,k = 0 (0 l q). By analogy with (7), we

set i,Ik = (

i,Il,k)0lq as the arg min in (0, . . . , q) of the quantity

E

YN,I,Itk+1 0 p0,k + hfk(

i1,Ik )

ql=1

l pl,kWl,k2

.(9)


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Iterating with i = 1, . . . , I , at the end we get (I,Il,k )0lq , thus, YN,I,Itk =I,I0,k p0,k and ZN,I,Il,tk =

I,Il,k pl,k (1 l q). The least squares problem (9)

can be formulated in different ways but this one is more convenient to getan intuition on (4). The error induced by this second step is analyzed by thefollowing result.

Theorem 2. Assume (H1)(H3). For h small enough, we have

max0kN

E|YN,I,Itk YNtk |2 + h

N1k=0

E|ZN,I,Itk ZNtk |2

Ch2I2

[1 + |S0|2

+ E|N

(PNtN)|

2

]

+ CN1k=0

E|Rp0,k(YNtk )|2 + ChN1k=0

ql=1

E|Rpl,k(ZNl,tk)|2.

The above result shows how projection errors cumulate along the back-ward iteration. The key point is to note that they only sum up, with afactor C which does not explode as N. These estimates improve thoseof Theorem 4.1 in [6] for two reasons. First, error estimates on ZN are pro-vided here. Second, in the cited theorem, the error is analyzed in terms ofE|Rp0,k(YN,I,Itk )|2 and E|Rpl,k(Z

N,I,Il,tk

)|2 say: hence, the influence of functionbases is still questionable, since it is hidden in the projection residuals

Rpk

and also in the random variables YN,I,Itk and ZN,I,Il,tk

. Our estimates are rel-

evant to directly analyze the influence of function bases (see Section 6 forexplicit computations). This feature is crucial in our opinion. Regarding theinfluence of I, it is enough here to have I= 2 to get an error of the sameorder as in Theorem 1. At the third step, I= 3 is needed.

Proof of Theorem 2. For convenience, we denote AN(S0) = 1 +|S0|2 +E|N(PNtN)|2. In the following computations, we repeatedly use threestandard inequalities:

1. The contraction property of the L2-projection operator: for any random

variable XL2, we have E|Ppl,k(X)|2

E|X|2

.2. The Young inequality: > 0, (a, b)R2, (a + b)2 (1 + h)a2 + ( 1 +

1h )b

2.

3. The discrete Gronwall lemma: for any nonnegative sequences (ak)0kN,(bk)0kN and (ck)0kN satisfying ak1+ ck1 (1 + h)ak + bk1 (with > 0), we have ak +

N1i=k ci e(Ttk)[aN +

N1i=k bi]. Most of the time,

it will be used with ci = 0.


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Step 4: upper bounds forN = h

N1k=0 E|ZN,I,Itk ZNtk |2. We aim at show-ing

N Ch2I2AN(S0) + ChN1k=0

ql=1

E|Rpl,k(ZNl,tk)|2

(26)

+ CN1k=0

E|Rp0,k(YNtk )|2 + C max

0kN1N,Ik .

In view of (23), we have

N

h

N1

k=0

q

l=1

E

|Rpl,k(Z

Nl,tk

)

|2

+ dN1k=0

(E[YN,I,Itk YNtk ]2 E[Ek(YN,I,Itk+1 YNtk+1)]2).

Owing to (21) and (24), we obtain

E|YN,I,Itk YNtk |2E[Ek(YN,I,Itk+1 YNtk+1 )]

2

Ch2I1AN(S0)+ CE|Rp0,k(YNtk )|2 + [(1+ h)(1 + h) 1]E|Ek[YN,I,Itk+1 YNtk+1 ]|

2

+ Chh + 1[E|Y

N,,Itk

YNtk |

2 + E|ZN,I,It

k ZNt

k |2].

Taking = 4Cd and h small enough such that dC(h + 1) 12 , we haveproved

NCh2I2AN(S0) + ChN1k=0

ql=1

E|Rpl,k(ZNl,tk)|2

+ CN1k=0

E|Rp0,k(YNtk )|2

+ C max0kN1

N,Ik +12h

N1k=0

E|YN,,Itk YNtk |2 + 12

N.

But taking into account (22) and (25) to estimate E|YN,,Itk YNtk |2, weclearly obtain (26). This easily completes the proof of Theorem 2.

5. Approximation results: step 3. This step is very analogous to step2, except that in the sequence of iterative least squares problems ( 9), theexpectation E is replaced by an empirical mean built on M independentsimulations of (PNtk )0kN, (Wk)0kN1. This leads to the algorithm thatis presented at Section 2.3. In this procedure, some truncation functions Nl,kand N,ml,k are used and we have to specify them now.


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These truncations come from a priori estimates on YN,i,Itk , ZN,i,Il,tk and it

is useful to force their simulation-based evaluations YN,i,I,M,mtk , ZN,i,I,M,ml,tk

to satisfy the same estimates. These a priori estimates are given by thefollowing result (which is proved later).

Proposition 2. Under (H1)(H3), for some constant C0 large enough,the sequence of functions (Nl,k() = max(1, C0|pl,k()|) : 0 l q, 0 k N1) is such that

|YN,i,Itk | N0,k(PNtk ),

h|ZN,i,Il,tk | Nl,k(PNtk ) a.s.,for any i

0, I

0 and 0

k

N

1.

With the notation of Section 2, the definition of the (random) truncation

functions Nl,k (resp. N,ml,k ) follows. Note that they are such that:

they leave invariant i,I0,k p0,k = YN,i,Itk if l = 0 or

hi,Il,k pl,k =

hZN,i,Il,tkif l 1 (resp. i,I0,k pm0,k if l = 0 or

h

i,Il,k pml,k if l 1);

they are bounded by 2Nl,k(PNtk ) [resp. 2Nl,k(PN,mtk )]; their first derivative is bounded by 1;

their second derivative is uniformly bounded in N,l,k,m.Now, we aim at quantifying the error between (YN,I,I,Mtk ,

hZN,I,I,Ml,tk )l,k and

(YN,I,I

tk,

hZN,I,I

l,tk)l,k , in terms of the number of simulations M, the function

bases and the time step h. The analysis here is more involved than in [6] sinceall the regression operators are correlated by the same set of simulated paths.To obtain more tractable theoretical estimates, we shall assume that eachfunction basis pl,k is orthonormal. Of course, this hypothesis does not affectthe numerical scheme, since the projection on a function basis is unchangedby any linear transformation of the basis. Moreover, we define the event

AMk = {j {k , . . . , N 1} :VMj Id h, PM0,j Id h

(27)and PMl,j Id 1 for 1 l q}

(see the notation of Section 2 for the definition of the matrices VMj and

PMl,j ). Under the orthonormality assumption for each basis pl,k, the matrices

(VMk )0kN1, (PMl,k )0lq,0kN1 converge to the identity with probabil-

ity 1 as M. Thus, we have limMP(AMk ) = 1. We now state ourmain result about the influence of the number of simulations.

Theorem 3. Assume (H1)(H3), I 3, that each function basis pl,k isorthonormal and thatE|pl,k|4


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for any 0 k N 1,

E|YN,I,Itk YN,I,I,Mtk

|2 + hN1j=k

E|ZN,I,Itj ZN,I,I,Mtj |2

9N1j=k

E(|Nj (PNtj )|21[AMk ]c) + ChI1

N1j=k

[1 + |S0|2 +E|Nj (PNtj )|2]

+C

hM

N1j=k

Evjvj Id2FE|Nj (PNtj )|2

+E(|vj|2

|p0,j+1

|2)E

|N

0,j

(PN

tj)|2

+ h2E

|vj |2(1 + |SNtj |2 + |p0,j |2E|N0,j(PNtj )|2

+1

h

ql=1

|pl,j |2E|Nl,j(PNtj )|2)

.

The term with [AMk ]c readily converges to 0 as M, but we have

not made estimations more explicit because the derivation of an optimalupper bound essentially depends on extra moment assumptions that maybe available. For instance, if Nj (P

Ntj ) has moments of order higher than 2,

we are reduced via Holder inequality to estimate the probability P([AMk ]

c

)N1j=k [P(VMj Id> h) +P(PM0,j Id> h) +

ql=1P(PMl,j Id > 1)]. We

have P(VMk Id > h) h2EVMk Id2 h2EVMk Id2F = (Mh2)1EvkvkId2F. This simple calculus illustrates the possible computations, other termscan be handled analogously.

The previous theorem is really informative since it provides a nonasymp-totic error estimation. With Theorems 1 and 2, it enables to see how tooptimally choose the time step h, the function bases and the number of sim-ulations to achieve a given accuracy. We do not report this analysis whichseems to be hard to derive for general function bases. This will be addressedin further researches [19]. However, our next numerical experiments give anidea of this optimal choice.

We conclude our theoretical analysis by stating a central limit theoremon the coefficients i,I,Mk as M goes to . This is less informative thanTheorem 3 since this is an asymptotic result. Thus, we remain vague aboutthe asymptotic variance. Explicit expressions can be derived from the proof.

Theorem 4. Assume (H1)(H3), that the driver is continuously dif-ferentiable w.r.t. (y, z) with a bounded and uniformly Holder continuous


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derivatives and that E|pl,k|2+ < for any k, l ( > 0). Then, the vector[

M(i,I,Mk i,Ik )]iI,kN1 weakly converges to a centered Gaussian vec-tor as M goes to .

Proof of Proposition 2. In view of Proposition 1, it is tempting toapply a Markov property argument and to assert that Proposition 2 resultsfrom (19) written with conditional expectations Ek. But this argumenta-tion fails because the law used for the projection is not the conditional lawEk but E0. The right argument may be the following one. Write Y

N,i,Itk

=

i,I0,k p0,k(PNtk ). On the one hand, by (19), we have CAN(S0) E|YN,i,Itk

|2 =i,I

0,k E[p0,kp

0,k

]i,I

0,k |i,I

0,k|2min(E[p0,kp

0,k

]). On the other hand,|YN,i,It

k | |i,I0,k||p0,k(PNtk )| |p0,k|

CAN(S0)/min(E[p0,kp0,k]). Thus, we can take N0,k(x) =max(1, |p0,k(x)|

CAN(S0)/min(E[p0,kp0,k])). Analogously, for

h|ZN,i,Il,tk |,

we have Nl,k(x) = max(1, |pl,k(x)|

CAN(S0)/min(E[pl,kpl,k])). Note that ifpl,k is an orthonormal function basis, we have min(E[pl,kp

l,k]) = 1 and pre-

vious upper bounds have simpler expressions.

Proof of Theorem 3. In the sequel, set

AN,Mk =1

M

M

m=1

|N0,k(PN,mtk )|2, BN,Mk =

1

M

M

m=1

|fmk (0, . . . , 0)|2.

Obviously, we have E(AN,Mk ) = E|N0,k(PNtk )|2 and E(BN,Mk ) C(1 + |S0|2).Now, we remind the standard contraction property in the case of leastsquares problems in RM, analogously to the case L2(,P). Consider a se-quence of real numbers (xm)1mM and a sequence (vm)1mM of vec-tors in Rn, associated to the matrix VM = 1M

Mm=1 v

m[vm] which is sup-posed to be invertible [min(V

M) > 0]. Then, the (unique) Rn-valued vectorx = arg inf |x v|2M is given by

x =[VM]1

M

M

m=1vmxm.(28)

The application x x is linear and, moreover, we have the inequalitymin(V

M)|x|2 |x v|2M |x|2M.(29)For the further computations, it is more convenient to deal with

(i,I,Mk ) = (i,I,M0,k

,

hi,I,M1,k

, . . . ,

hi,I,Mq,k

)


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instead of i,I,Mk . Then, the Picard iterations given in (4) can be rewritten

i+1,I,Mk = arg inf

1

M

Mm=1

(N,m0,k+1(I,I,M0,k+1 pm0,k+1) + hfmk (i,I,Mk ) vmk )2.(30)

Introducing the event AMk , taking into account the Lipschitz property of thefunctions Nl,k and using the orthonormality of pl,k, we get

E|YN,I,Itk YN,I,I,Mtk

|2 + hN1j=k

E|ZN,I,Itj ZN,I,I,Mtj |2

9

N1

j=k

E(|N

j

(PN

tj)|21

[AM

k ]c)(31)

+E(1AMk|I,I,M0,k I,I0,k|2) + h

N1j=k

ql=1

E(1AMk|I,I,Ml,j I,Il,j |2).

To obtain Theorem 3, we estimate |I,I,Mk I,Ik |2 on the event AMk . This isachieved in several steps.

Step 1: contraction properties relative to the sequence (i,I,Mk )i0. Theyare summed up in the following lemma:

Lemma 1. For h small enough, onAMk the following properties hold:

(a) |i+1,I,Mk i,I,Mk |2 Ch|i,I,Mk i1,I,Mk |2.(b) There is a unique vector ,I,Mk such that

,I,Mk = arg inf

1

M

Mm=1

(N,m0,k+1(I,I,M0,k+1 pm0,k+1) + hfmk (,I,Mk ) vmk )2.

(c) We have |,I,Mk I,I,Mk |2 [Ch]I|,I,Mk |2.

Proof. We prove (a). Since 1 h min(VMk ) and max(PMl,k ) 2 (0 l q) on AMk , in view of (29), we obtain that (1 h)|i+1,I,Mk i,I,Mk |2 isbounded by

h2

M

Mm=1

(fmk (i,I,Mk ) fmk (i1,I,Mk ))2

Ch2q

l=0

|i,I,Ml,k i1,I,Ml,k |2max(PMl,k )

Ch|i,I,Mk i1,I,Mk |2.


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Now, statements (a) and (b) are clear. For (c), apply (a), reminding that0,I,Mk = 0.

Step 2: bounds for |i,I,Mk | on the eventAMk . Namely, we aim at showingthat

|i,I,Mk |2 C(AN,Mk+1 + hBN,Mk ) on AMk .(32)We first consider i =. As in the proof of Lemma 1, we get

(1 h)|,I,Mk |2

1

M

M

m=1

[N,m0,k+1(I,I,M0,k+1

pm0,k+1) + hf

mk (

,I,Mk )]

2

(1 + h)AN,Mk+1 + Ch

h +1

BN,Mk +

ql=0

|,I,Ml,k |2max(PMl,k )

.

Take = 8C and h small enough to ensure 2C(h + 1)(1 + h) 12(1 h). Itreadily follows |,I,Mk |2 C(AN,Mk+1 + hBN,Mk ), proving that (32) holds fori =. Lemma 1(c) leads to expected bounds for other values of i.

Step 3: we remind bounds for i,I. Using Proposition 2 and in view of(10)(14), we have, for i 1,

|i,I

l,k |2

E

|N

l,k(P

N

tk )|2

, 0 l q;(33)|,Ik i,Ik |2 (Cfh)2iE|N0,k(PNtk )|2.

Remember also the following expression of ,Ik , derived from (10)(13) andthe orthonormality of each basis pl,k:

,Ik = E(vk[I,I0,k+1 p0,k+1 + hfk(,Ik )]).(34)

Step 4: decomposition of the quantity E(1AMk|I,I,Mk I,Ik |2). Due to

Lemma 1, on AMk we get |,I,Mk I,I,Mk |2 ChI|,I,Mk |2 ChI|,Ik |2 +ChI|,I,Mk ,Ik |2. Thus, using (33), it readily follows that E(1AM

k|I,I,Mk

I,Ik |2) is bounded by(1 + h)E(1

AMk|,I,Mk ,Ik |2)

+ 2

1 +

1

h

{E(1

AMk|I,I,Mk ,I,Mk |2) + |I,Ik ,Ik |2}(35)

(1 + Ch)E(1AMk|,I,Mk ,Ik |2) + ChI1E|Nk (PNtk )|2,


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taking account that I 3. On AMk , VMk is invertible and we can setB1 = (Id (VMk )1),Ik ,

B2 = (VMk )

1E(vk

N0,k+1(

I,I0,k+1 p0,k+1))

1

M

Mm=1

vmk N,m0,k+1(

I,I0,k+1 pm0,k+1)

,

B3 = (VMk )

1h

E(vkfk(

,Ik ))

1

M

Mm=1

vmk fmk (

,Ik )

,

B4 =(VMk )

1

M

Mm=1

vmk [N,m0,k+1(

I,I0,k+1 pm0,k+1) N,m0,k+1(I,I,M0,k+1 pm0,k+1)

+ h(fmk (,Ik ) fmk (,I,Mk ))].

Thus, by (28)(34), we can write ,Ik ,I,Mk = B1 + B2 + B3 + B4, whichgives on AMk

|,Ik ,I,Mk |2 3

1 +1

h

(|B1|2 + |B2|2 + |B3|2) + ( 1 + h)|B4|2.(36)

Step 5: individual estimation of B1, B2, B3, B4 on AMk . Remember

the classic result [16]: ifIdF< 1, F1 Id =k=1[IdF]k and IdF1 FId1FId . Consequently, for F = VMk , we get E(1AMk Id(V

Mk )

12)(1 h)2EId VMk 2 (1 h)2EVMk Id2F = (M(1 h)2)1Evkvk Id

2

F. Thus, we haveE(|B1|21AM

k) C

MEvkvk Id2FE|Nk (PNtk )|

2.

Since on AMk one has (VMk )1 2, it readily followsE(|B2|21AM

k) C

ME(|vk|2|p0,k+1|2)E|N0,k(PNtk )|2,

E(|B3|21AMk

) Ch2

ME

|vk|2

1 + |SNtk |

2 + |p0,k|2E|N0,k(PNtk )|2

+1

h

q

l=1

|pl,k|2E|Nl,k(PNtk )|2

.

As in the proof of Lemma 1 and using PM0,k+1 1 + h on AMk , we easilyobtain

(1 h)|B4|2 (1 + h)(1 + h)|I,I0,k+1 I,I,M0,k+1|2

+ Ch

h +

1

ql=0

|,Il,k ,I,Ml,k |2.


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Step 6: final estimations. Put k = EvkvkId2FE|Nk (PNtk )|2+E(|vk|2|p0,k+1|2)E|N0,k(PNtk )|2h2E[|vk|2(1 + |SNtk |2 + |p0,k|2E|N0,k(PNtk )|2 + 1h

ql=1 |pl,k|2E|Nl,k(PNtk )|2)]. Plug

the above estimates on B1, B2, B3, B4 into (36), choose = 3C and h closeto 0 to ensure Ch + C 12 ; after simplifications, we get

E(1AMk|,I,Mk ,Ik |2)C

khM

+ ( 1 + Ch)E(1AMk|I,I0,k+1I,I,M0,k+1|2).

But in view of Lemma 1(c) and estimates (32)(33), we have

E(1AMk|I,I0,k+1I,I,M0,k+1|2)

(1 + h)E(1AMk|,I0,k+1,I,M0,k+1 |2)

+ ChI1(1 + |S0|2 + E|N0,k+1(PNtk+1 )|2 + E|N0,k+2(PNtk+2 )|

2).

Finally, we have proved

E(1AMk|,I,Mk ,Ik |2)

C khM

+ ChI1(1 + |S0|2 +E|N0,k+1(PNtk+1 )|2 + E|N0,k+2(PNtk+2 )|

2)

+ ( 1 + Ch)E(1AMk|,I,M0,k+1 ,I0,k+1|2).

Using a contraction argument as in (35), the index can be replaced byI, without changing the inequality (with a possibly different constant C).

This can be written

E(1AMk|I,I,M0,k I,I0,k|2) + h

ql=1

E(1AMk|I,I,Ml,k I,Il,k |2)

C khM

+ ChI1(1 + |S0|2 +E|N0,k+1(PNtk+1 )|2 + E|N0,k+2(PNtk+2 )|

2)

+ ( 1 + Ch)E(1AMk|I,I,M0,k+1 I,I0,k+1|2).

Using Gronwalls lemma, the proof is complete.

Remark 1. The attentive reader may have noted that powers of h aresmaller here than in Theorem 2, which leads to take I

3 instead of I

2

before. Indeed, we cannot take advantage of conditional expectations on thesimulations as we did in (12), for instance.

Note that in the proof above, we only use the Lipschitz property of thetruncation functions Nl,k and

N,ml,k .

Proof of Theorem 4. The arguments are standard and there areessentially notational difficulties. The first partial derivatives of f w.r.t. y


23/31


and zl are, respectively, denoted 0f and lf. The parameter ]0, 1] standsfor their Holder continuity index. Suppose w.l.o.g. that < and that eachfunction basis pl,k is orthonormal. For k < N 1, define the quantities

AMl,k() =1

M

Mm=1

vmk lf(tk, SN,mtk

, 0 pm0,k, . . . , q pmq,k)[pml,k],

BMk =1

M

Mm=1

vmk [pm0,k+1]

, DMk =

M(Id VMk ),

CMk ()

=M

m=1

{vmk [I,I0,k+1 pm0,k+1 + hfmk ()]E(vk[I,I0,k+1 p0,k+1 + hfk()])}M

.

For k = N 1, we set BMk = 0 and in CMk (), the terms I,I0,k+1 pm0,k+1 andI,I0,k+1 p0,k+1 have to be replaced, respectively, by N(PN,mtN ) and N(PNtN).The definitions of AMl,k() and D

Mk are still valid. For convenience, we write

XMw if the (possibly vector or matrix valued) sequence (XM)M weakly

converges to a centered Gaussian variable, as M goes to infinity. For the con-

vergence in probability to a constant, we denote XMP. Since simulations

are independent, observe that the following convergences hold:

(AMl,k(i,Ik ), B

Mk , V

Mk )iI1,lq,kN1

P ,(37)

GM = (CMk (i,Ik ), DMk )iI1,lq,kN1 w .

Note that limM VMka.s.= Id is invertible. Linearizing the functions f and

N,m0,k+1 in the expressions ofi,Ik = E(vk[

I,I0,k+1 p0,k+1+hfk(i1,I0,k , . . . , i1,Iq,k )])

and i,I,Mk given by (28) leads toVMk

M(i,I,Mk i,Ik )DMk i,Ik CMk (i1,Ik )

BMk M(I,I,M0,k+1 I,I0,k+1)

hq

l=0

AMl,k(i1,Ik )

M(

i1,I,Ml,k i1,Il,k )

(38) 1k


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+CM|i1,I,Mk i1,Ik |1+

Mm=1

|vmk ||pmk |1+.

To get Theorem 4, we prove by induction on k that ([

M(i,I,Mj i,Ij )]jk,iI,GM) w.Remember that 0,I,Mj =

0,Ij = 0 for any j. Consider first k = N 1, for

which BMk = 0, and i = 1. In view of (37)(38), clearly ([

M(i,I,MN1 i,IN1)]i1,GM) w.For i = 2, we may invoke the same argument using (37)(38) and obtain

([

M(i,I,MN1 i,IN1)]i2,GM) w provided that the upper bound in (38) con-verge to 0 in probability. To prove this, put MM = M1/2Mm=1 |vmN1||pmN1|1+and write 1

M|1,I,MN1 1,IN1|1+

Mm=1 |vmN1| |pmN1|1+ = |

M(1,I,MN1

1,I

N1)|1+

MM

. Since [M(1,I,M

N1 1,I

N1)]M is tight, our assertion holds ifMM converges to 0 as M. Note that |vN1||pN1|1+ L(2+)/(2+)(P).Thus, the strong law of large numbers, in the case of i.i.d. random variableswith infinite mean, leads to

Mm=1 |vmN1| |pmN1|1+ = O(M(2+)/(2+)+r)

a.s. for any r > 0. Consequently, from the choice of r small enough, it followsMM 0 a.s.

Iterating this argumentation readily leads to ([

M(i,I,MN1 i,IN1)]iI,GM) w.For the induction for k < N 1, we apply the techniques above. There is anadditional contribution due to BMk , which can be handled as before.

6. Numerical experiments.

6.1. Lipschitz property of the solution under (H4). To use the algorithm,we need to specify the basis functions that we choose at each time tk and forthis, the knowledge of the regularity of the functions yNk () and zNl,k() fromProposition 1 is useful (in view of Theorem 2). In all the cases describedin Section 2.4 and below, assumption (H4) is fulfilled. Under this extraassumption, we now establish that yNk () and zNl,k() are Lipschitz continuous.

Proposition 3. Assume (H1)(H4). For h small enough, we have

|yNk0 (x) yNk0(x)|+

h|zNk0 (x) zNk0 (x)| C|x x|(39)uniformly in k0 N 1.

Proof. As for (17), we can obtainE|YN,k0,xtk Y

N,k0,x

tk|2

(1 + h)1Ch(h + 1/)E|Ek(Y

N,k0,xtk+1

YN,k0,xtk+1 )|2

+Ch(h + 1/)

1Ch(h + 1/)E|SN,k0,xtk

SN,k0,xtk |2


25/31


+ C(h + 1/)1Ch(h + 1/) (E|Y

N,k0,xtk+1

YN,k0,xtk+1 |2

E|Ek(YN,k0,xtk+1 YN,k0,x

tk+1)|2).

Choosing = C and h small enough, we get (for another constant C)

E|YN,k0,xtk YN,k0,x

tk |2

(1 + Ch)E|YN,k0,xtk+1 YN,k0,x

tk+1|2 + ChE|SN,k0,xtk S

N,k0,x

tk|2.

The last term above is bounded by C|xx|2 under assumption (H1). Thus,using Gronwalls lemma and assumption (H4), we get the result for yNk0 ().

The result for

hz

N

k0 () follows by considering (5).

6.2. Choice of function bases. Now, we specify several choices of functionbases. We denote d ( d) the dimension of the state space of (PNtk )k.

Hypercubes (HC in the following). Here, to simplify, pl,k does not de-

pend on l or k. Choose a domain D Rd centered on PN0 , that is, D =di=1 ]P

N0,i R, PN0,i + R], and partition it into small hypercubes of edge .

Thus, D =

i1,...,idDi1,...,id where Di1,...,id = ]P

N0,1R + i1, PN0,1R + (i1 +

1)] ]PN0,dR+id, PN0,dR+(id +1)]. Then we define pl,k as the in-dicator functions associated to this set of hypercubes:pl,k() = (1Di1,...,id ())i1,...,id .With this particular choice of function bases, we can explicit the projectionerror of Theorem 2:

E(Rp0,k(YNtk )2)

E(|YNtk |21Dc(P

Ntk

)) +

i1,...,id

E(1Di1,...,id(PNtk )|yNk (PNtk )yNk (xi1,...,id )|

2)

C2 +E(|YNtk |21Dc(PNtk )),where xi1,...,id is an arbitrary point of Di1,...,id and where we have used the

Lipschitz property of yNk on D. To evaluate E(|YNtk |21Dc(PNtk )), note that,by adapting the proof of Proposition 3, we have |YNtk |2 C(1 + |SNtk |2 +Ek|PNtN|2). Thus, if supk,NE|PNtk | 2, we have E(|YNk |21Dc(PNtk ))

CR2

, with an explicit constant C. The choice R

h2/(2) and = h leads

to

E|Rp0,k(YNtk )|2 Ch2.The same estimates hold for E|Rpl,k(

hZNl,tk)|2. Thus, we obtain the same

accuracy as in Theorem 1.Voronoi partition (VP). Here, we consider again a basis of indicator func-

tions and the same basis for all 0 l q. This time, the sets of the indicator


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functions are an open Voronoi partition ([17]) whose centers are indepen-dent simulations of PN. More precisely, if we want a basis of 20 indicatorfunctions, we simulate 20 extra paths of PN, denoted (PN,M+i)1i20, in-dependently of (PN,m)1mM. Then we take at time tk (P

N,M+itk

)1i20 todefine our Voronoi partition (Ck,i)1i20, where Ck,i = {x : |x PN,M+itk |

re gresi on based montecarlo methods

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