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On pathwise stochastic integration with finite variationprocesses uniformly approximating cadlag processes
Rafa l Marcin Lochowski
Warsaw School of Economics
Torun 2013
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 1 / 23
Truncated variation - how it appears
Let f : [a; b]→ R be a cadlag function and let c > 0Question: what is the smallest total variation possible of a (cadlag)function from the ball
{g : ‖f − g‖∞ ≤
12c}
?
The immediate bound frombelow reads as
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) ≥ TV c (f , [a; b]) ,
where
TV (g , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
|g (ti )− g (ti−1)| ,
TV c (f , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
and follows immediately from the inequality
|g (ti )− g (ti−1)| ≥ max {|f (ti )− f (ti−1)| − c , 0} .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 2 / 23
Truncated variation - how it appears
Let f : [a; b]→ R be a cadlag function and let c > 0Question: what is the smallest total variation possible of a (cadlag)function from the ball
{g : ‖f − g‖∞ ≤
12c}
? The immediate bound frombelow reads as
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) ≥ TV c (f , [a; b]) ,
where
TV (g , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
|g (ti )− g (ti−1)| ,
TV c (f , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
and follows immediately from the inequality
|g (ti )− g (ti−1)| ≥ max {|f (ti )− f (ti−1)| − c , 0} .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 2 / 23
Truncated variation - how it appears
Let f : [a; b]→ R be a cadlag function and let c > 0Question: what is the smallest total variation possible of a (cadlag)function from the ball
{g : ‖f − g‖∞ ≤
12c}
? The immediate bound frombelow reads as
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) ≥ TV c (f , [a; b]) ,
where
TV (g , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
|g (ti )− g (ti−1)| ,
TV c (f , [a; b]) := supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
and follows immediately from the inequality
|g (ti )− g (ti−1)| ≥ max {|f (ti )− f (ti−1)| − c , 0} .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 2 / 23
Truncated variation - definition and another interpretation
We will call the quantity
TV c (f , [a; b]) = supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
truncated variation of the function f at the level c .
Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .It also possible to show that in fact we have the equality
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) = TV c (f , [a; b]) .
For f : [0; +∞)→ R we will denote
TV c (f , [0; t]) =: TV c (f , t) .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 3 / 23
Truncated variation - definition and another interpretation
We will call the quantity
TV c (f , [a; b]) = supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
truncated variation of the function f at the level c .Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .
It also possible to show that in fact we have the equality
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) = TV c (f , [a; b]) .
For f : [0; +∞)→ R we will denote
TV c (f , [0; t]) =: TV c (f , t) .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 3 / 23
Truncated variation - definition and another interpretation
We will call the quantity
TV c (f , [a; b]) = supn
supa≤t0<t1<...<tn≤b
n∑i=1
max {|f (ti )− f (ti−1)| − c , 0} ,
truncated variation of the function f at the level c .Trunated variation may be interpreted not only as the lower boundobtained on the previous slide but also as the variation taking into accountonly jumps greater than c .It also possible to show that in fact we have the equality
infg :‖f−g‖∞≤
12 c
TV (g , [a; b]) = TV c (f , [a; b]) .
For f : [0; +∞)→ R we will denote
TV c (f , [0; t]) =: TV c (f , t) .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 3 / 23
Assymptotic properties of truncated variation
It appears that the truncated variation is closely related with p−variation,defiend as
V p (f , t) = supn
sup0≤t0<t1<...<tn≤t
n∑i=1
|f (ti )− f (ti−1)|p .
Let
Vp be the class of functions f : [0; +∞)→ R with locally finitep−variation and
Up be the class of such functions f that for any t > 0,
lim supc↓0
cp−1TV c (f , t) ∈ (0; +∞)
In [5] it is shown that for any p ≥ 1 and δ > 0 we have inclusionsVp ⊂ Up ⊂ Vp+δ and for p > 1 these inclusions are strict.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 4 / 23
Limit distributions of truncated variation processes ofBrownian motion with drift as c → 0
Let Xt = Bt , t ≥ 0, be a standard Brownian motion. Since it has infinitetotal variation on any interval [0; t], t > 0, for any t > 0
limc↓0
TV c (X , t) =∞.
It may be of interest (due to the geometric interpretation of truncatedvariation and its assymptotic properties) to investigate the rate ofTV c (X , t) for small cs. The answer is the following
Theorem ([4])
For any T > 0 the process c · TV c (X , t) converges almost surely in(C[0;T ],R) topology to the deterministic functionid : [0;T ]→ R, id(t) = t.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 5 / 23
Limit distributions of truncated variation processes ofBrownian motion with drift as c → 0
Let Xt = Bt , t ≥ 0, be a standard Brownian motion. Since it has infinitetotal variation on any interval [0; t], t > 0, for any t > 0
limc↓0
TV c (X , t) =∞.
It may be of interest (due to the geometric interpretation of truncatedvariation and its assymptotic properties) to investigate the rate ofTV c (X , t) for small cs. The answer is the following
Theorem ([4])
For any T > 0 the process c · TV c (X , t) converges almost surely in(C[0;T ],R) topology to the deterministic functionid : [0;T ]→ R, id(t) = t.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 5 / 23
Generalisation for continuous semimartingales
Let now Xt , t ≥ 0, be continuous a semimartingale, with thedecomposition Xt = X0 + Mt + At , with Mt being a local martingale andAt being a continuous process with finite total variation.Using the inequality max {|x + y | − c , 0} ≤ max {|x | − c , 0}+ |y | weobtain, that
TV c (Xt , t) ≤ TV c (Mt , t) + TV 0 (At , t)
andTV c (Mt , t) ≤ TV c (Xt , t) + TV 0 (−At , t) .
Hence we conclude easily that
limc↓0
c · TV c (X , t) = limc↓0
c · TV c (M, t)
whenever any of the above limits exists.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 6 / 23
Generalisation for continuous semimartingales, cont.
We will use the Theorem from the previous slide, Dambis andDubins-Schwarz Theorem saying that every continuous, local martingaleMt , t ≥ 0, with M0 = 0 and infinite total variation may be represented as
Mt = B〈M,M〉t ,
where Bt is a standard Brownian motion and the fact that the truncatedvariation does not depend on continuous and strictly increasing change oftime variable. Utilizing above facts, we obtain that
limc↓0
c ·TV c (X , t) = limc↓0
c·TV c (M, t) = limc↓0
c ·TV c (B, 〈M,M〉t) = 〈M,M〉t .
Noticing that 〈X ,X 〉t = 〈M,M〉t , we finally obtain
limc↓0
c · TV c (X , t) = 〈X ,X 〉t .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 7 / 23
Concentration properties
Truncated variation seems to be more informative than p−variation:
the finitness of p− variation may be recovered form the assymptoticproperties of the truncated variation;
for fixed c > 0, one may look at TV c as a random variable with thenatural geometric interpretation mentioned.
It appears that for fixed c > 0 and for X being a fBm or a diffusion withmoderate growth, TV c reveals strong concentration properties. Forexample, for a standard Brownian motion we have
Theorem ([1])
For the standard Brownian motion X = B, t · c−1 is comparable up to auniversal constant with ETV c (X , t) and for some universal constants A,Bthe Gaussian concentration holds
P(TV c(B, t) ≥ A · t · c−1 + B√tu) ≤ exp(−u2), for u ≥ 0.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 8 / 23
Concentration properties
Truncated variation seems to be more informative than p−variation:
the finitness of p− variation may be recovered form the assymptoticproperties of the truncated variation;
for fixed c > 0, one may look at TV c as a random variable with thenatural geometric interpretation mentioned.
It appears that for fixed c > 0 and for X being a fBm or a diffusion withmoderate growth, TV c reveals strong concentration properties. Forexample, for a standard Brownian motion we have
Theorem ([1])
For the standard Brownian motion X = B, t · c−1 is comparable up to auniversal constant with ETV c (X , t) and for some universal constants A,Bthe Gaussian concentration holds
P(TV c(B, t) ≥ A · t · c−1 + B√tu) ≤ exp(−u2), for u ≥ 0.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 8 / 23
The Skorohod problem
The truncated variation also appears to be related with the Skorohodproblem on [−c/2; c/2].
Let
D[0; +∞) be a set of real-valued, cadlag functions, defined on theinterval [0; +∞),
BV [0; +∞) denote a subset of D[0; +∞) consisting of functionswith locally finite total variation and
I [0; +∞) denote a subset of D[0; +∞) consisting of non-decreasingfunctions.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 9 / 23
The Skorohod problem
The truncated variation also appears to be related with the Skorohodproblem on [−c/2; c/2]. Let
D[0; +∞) be a set of real-valued, cadlag functions, defined on theinterval [0; +∞),
BV [0; +∞) denote a subset of D[0; +∞) consisting of functionswith locally finite total variation and
I [0; +∞) denote a subset of D[0; +∞) consisting of non-decreasingfunctions.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 9 / 23
The Skorohod problem, cont.
A pair of functions (φ,−ξ) ∈ D[0; +∞)× BV [0; +∞) is said to be asolution of the Skorohod problem on [−c/2, c/2] with starting conditionξ(0) = ξ0 for u ∈ D[0; +∞) if the following conditions are satisfied:
1 for every t ≥ 0, φ (t) = u (t)− ξ (t) ∈ [−c/2, c/2] ;
2 ξ = ξu − ξd , where ξu, ξd ∈ I [0; +∞) and the corresponding measuresdξu, dξd are carried by {t ≥ 0 : φ (t) = c/2} and{t ≥ 0 : φ (t) = −c/2} respectively;
3 ξ(0) = ξ0.
For ξ0 ∈ [u (0)− c/2; u (0) + c/2] the Skorohod problem has a uniquesolution.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 10 / 23
Graphical interpretation of the Skorohod problem
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 11 / 23
The Skorohod problem and the truncated variation
Let uc,ξ0
be the solution of the Skorohod problem with starting conditionξ0 ∈ [u(0)− c/2; u(0) + c/2].For any such ξ0 and t > 0 we have
TV c(u, t) ≤ TV(uc,ξ
0, t)≤ TV c(u, t) + c .
For simplicity let us set uc = uc,u(0). From the properties of the Skorohodmap, we get∫ t
0(u − uc)duc =
∫ t
0
c
2ducu −
∫ t
0−c
2ducd =
c
2· TV (uc , t) ,
where ucu and ucd are non-decreasing functions from the definition of theSkorohod problem, such that uc = uc(0) + ucu − ucd .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 12 / 23
The Skorohod problem and the truncated variation
Let uc,ξ0
be the solution of the Skorohod problem with starting conditionξ0 ∈ [u(0)− c/2; u(0) + c/2].For any such ξ0 and t > 0 we have
TV c(u, t) ≤ TV(uc,ξ
0, t)≤ TV c(u, t) + c .
For simplicity let us set uc = uc,u(0). From the properties of the Skorohodmap, we get∫ t
0(u − uc)duc =
∫ t
0
c
2ducu −
∫ t
0−c
2ducd =
c
2· TV (uc , t) ,
where ucu and ucd are non-decreasing functions from the definition of theSkorohod problem, such that uc = uc(0) + ucu − ucd .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 12 / 23
Wong-Zakai’s pathwise approach to the stochastic integral
Since many years probabilists tried to define the stochastic integral in apathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:
(A) for all t ∈ [0;T ], Bnt → Bt pointwise as n ↑ +∞, where Bn,
n = 1, 2, . . . , are continuous and have locally bounded variation;
(B) (A) and there exists such a locally bounded process Z that for allt ∈ [0;T ], |Bn
t | ≤ Zt ;
and stated the following
Theorem ([6])
Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ
∂x and let Bn satisfy
(B), then for the Lebesgue-Stieltjes integrals∫ T
0 ψ (t,Bnt )dBn
t , a.s.,
limn→∞
∫ T
0ψ (t,Bn
t )dBnt =
∫ T
0ψ (t,Bt)dBt +
1
2
∫ T
0
∂ψ
∂x(t,Bt)dt.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 13 / 23
Wong-Zakai’s pathwise approach to the stochastic integral
Since many years probabilists tried to define the stochastic integral in apathwise way.One of the earliest of such attemps is due to Wong and Zakai (1965). ForT > 0 they considered the following approximation of Brownian paths:
(A) for all t ∈ [0;T ], Bnt → Bt pointwise as n ↑ +∞, where Bn,
n = 1, 2, . . . , are continuous and have locally bounded variation;
(B) (A) and there exists such a locally bounded process Z that for allt ∈ [0;T ], |Bn
t | ≤ Zt ;
and stated the following
Theorem ([6])
Let ψ(t, x) has continuous partial derivatives ∂ψ∂t and ∂ψ
∂x and let Bn satisfy
(B), then for the Lebesgue-Stieltjes integrals∫ T
0 ψ (t,Bnt )dBn
t , a.s.,
limn→∞
∫ T
0ψ (t,Bn
t )dBnt =
∫ T
0ψ (t,Bt)dBt +
1
2
∫ T
0
∂ψ
∂x(t,Bt) dt.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 13 / 23
Disadvantages of the Wong-Zakai construction
The Wong-Zakai construction can not be extended when oneconsiders different approximating sequences of the underlyingBrownian motion in integrator and integrand.
Using properties of the Skorohod map and the truncated variation itis relatively easy to construct an appropriate example.
Set: Y n := B1/n2+ n
(B1/(2n2) − B1/n2
). We easily check that Y n and
Zn := B1/n2satisfy (A)-(B) for B. We have Bc/2 − Bc ≥ c/4 on the set
dBc > 0 and Bc/2 − Bc ≤ −c/4 on the set dBc < 0. Thus∫ 1
0Y ndZn −
∫ 1
0ZndZn =
∫ 1
0n(B1/(2n2) − B1/n2
)dB1/n2
≥ n1
4n2
∫ 1
0
∣∣∣dB1/n2∣∣∣ =
n
4n−2TV
(B1/n2
, 1)≥ n
4n−2TV 1/n2
(B, 1) .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 14 / 23
Disadvantages of the Wong-Zakai construction
The Wong-Zakai construction can not be extended when oneconsiders different approximating sequences of the underlyingBrownian motion in integrator and integrand.
Using properties of the Skorohod map and the truncated variation itis relatively easy to construct an appropriate example.
Set: Y n := B1/n2+ n
(B1/(2n2) − B1/n2
). We easily check that Y n and
Zn := B1/n2satisfy (A)-(B) for B. We have Bc/2 − Bc ≥ c/4 on the set
dBc > 0 and Bc/2 − Bc ≤ −c/4 on the set dBc < 0. Thus∫ 1
0Y ndZn −
∫ 1
0ZndZn =
∫ 1
0n(B1/(2n2) − B1/n2
)dB1/n2
≥ n1
4n2
∫ 1
0
∣∣∣dB1/n2∣∣∣ =
n
4n−2TV
(B1/n2
, 1)≥ n
4n−2TV 1/n2
(B, 1) .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 14 / 23
The Skorohod problem approximating sequence
Let Xt , t ≥ 0, be a process with cadlag paths. The process X c obtainedvia the Skorohod map has the following properties:
(i) X c has locally finite total variation;
(ii) X c has cadlag paths;
(iii) for every T ≥ 0
|Xt − X ct | ≤
1
2c;
(iv) for every T ≥ 0|∆X c
t | ≤ |∆Xt | ,
where ∆X ct = X c
t − X ct−, ∆Xt = Xt − Xt−;
(v) the process X c is adapted to the natural filtration of X .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 15 / 23
A generalisation of the Skorohod problem approximatingsequence
In [3] for any cadlag process X the small generalisation is considered. Forany c > 0 we consider a process X c such that
(i) X c has locally finite total variation;
(ii) X c has cadlag paths;
(iii) for every T ≥ 0 there exists such KT < +∞ that for every t ∈ [0;T ] ,
|Xt − X ct | ≤ KT c ;
(iv) for every T ≥ 0 there exists such LT < +∞ that for every t ∈ [0;T ] ,
|∆X ct | ≤ LT |∆Xt | ,
where ∆X ct = X c
t − X ct−, ∆Xt = Xt − Xt−;
(v) the process X c is adapted to the natural filtration of X .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 16 / 23
A modification of the Wong-Zakai construction, cont.
Further, in [3] the following theorems were proven.
Theorem
If processes X and Y are cadlag semimartingales then for the sequence ofthe pathwise Lebesgue-Stieltjes integrals
∫ T0 Y−dX
c we have∫ T
0Y−dX
c →Pc↓0
∫ T
0Y−dX + [X ,Y ]contT .
∫ T0 Y−dX denotes here the (semimartingale) stochastic integral and
[X ,Y ]cont denotes here the continuous part of [X ,Y ], i.e.
[X ,Y ]contT = [X ,Y ]T −∑
0<s≤T∆Xs∆Ys .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 17 / 23
A modification of the Wong-Zakai construction, cont.
Moreover
Theorem
When c(n) > 0 and∑+∞
n=1 c(n)2 < +∞ then we have∫ T
0Y−dX
c(n) →∫ T
0Y−dX + [X ,Y ]contT a.s.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 18 / 23
Drawbacks of the construction presented
Unfortunately, the construction presented does not work for any cagladintegrand Y .It is possible to construct a continuous, globally bounded, adapted tothe natural Brownian filtration process Y and a sequence Bc(n),n = 1, 2, . . . , satisfying all conditions (i)-(v) for X = B such that theintegral ∫ 1
0Y dBc(n)
diverges.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 19 / 23
Drawbacks of the construction presented, cont.
First (cf. [3]) we define sequence b (n) , n = 1, 2, . . . in the following wayb (1) = 1 and for n = 2, 3, . . .
b (n) = n2b (n − 1)6 .
Now we define a (n) := b (n)1/2 , c (n) := b (n)−1 and set
Y :=∞∑n=2
a (n)(B − Bc(n)
).
The proof also utilises the concentration properties of TV c .
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 20 / 23
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤t≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧t
(Xτni ∧t − Xτni−1∧t
)−∫ t
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ 2−n}.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ c (n)}.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 21 / 23
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤t≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧t
(Xτni ∧t − Xτni−1∧t
)−∫ t
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ 2−n}.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ c (n)}.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 21 / 23
Bichteler’s construction
The remarkable Bichteler’s approach provides pathwise construction forintegration of any adapted cadlag process Y with cadlag semimartingaleintegrator X and is based on the approximation
limn→∞
sup0≤t≤T
∣∣∣∣∣Y0X0 +∞∑i=1
Yτni−1∧t
(Xτni ∧t − Xτni−1∧t
)−∫ t
0Y−dX
∣∣∣∣∣ = 0 a.s.,
where τn = (τni ) , i = 0, 1, 2, . . . , is the following sequence of stoppingtimes: τn0 = 0 and for i = 1, 2, . . . ,
τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ 2−n}.
Remark
In fact, given c(n) > 0,∑∞
n=1 c2 (n) < +∞, Bichteler’s construction works
for any sequence τn = (τni ) , i = 0, 1, 2, . . . , of stopping times, such that
τn0 = 0 and for i = 1, 2, . . . , τni = inf{s > τni−1 :
∣∣∣Ys − Yτni−1
∣∣∣ ≥ c (n)}.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 21 / 23
Some references
Bednorz, W., Lochowski, R. M., (2012) Integrability and concentration ofsample paths’ truncated variation of fractional Brownian motions, diffusionsand Levy processes. Submitted to Bernoulli, revision requested.
Bichteler, K., (1981) Stochastic integration and Lp− theory ofsemimartingales. Ann. Probab., 9(1):49–89.
Lochowski, R., M., (2013) Pathwise stochatic integration with finitevariation processes uniformly approximating cadlag processes, submitted.
Lochowski, R. M. and Mi los, P., (2013) On truncated variation, upwardtruncated variation and downward truncated variation for diffusions.Stochastic Process. Appl., 123(2):446–474.
Tronel, G. and Vladimirov, A. A., (2000) On BV-type hysteresis operators.Nonlinear Anal., 39(1), Ser. A: Theory Methods, 79–98.
Wong, E. and Zakai, M., (1965) On the convergence of ordinary integrals tostochastic integrals. Ann. Math. Statist., 36:1560–1564.
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 22 / 23
Thank you!
Rafa l Marcin Lochowski (WSE) On pathwise stochastic integration Torun 2013 23 / 23