On pathwise stochastic integration with finite variation ...rlocho/PathStochInt2.pdf · On pathwise...

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


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} .


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


12 c

TV (g , [a; b]) ≥ TV c (f , [a; b]) ,

where

TV (g , [a; b]) := supn


n∑i=1

|g (ti )− g (ti−1)| ,

TV c (f , [a; b]) := supn


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} .


Truncated variation - definition and another interpretation

We will call the quantity

TV c (f , [a; b]) = supn


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


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) .






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


12 c

TV (g , [a; b]) = TV c (f , [a; b]) .


TV c (f , [0; t]) =: TV c (f , t) .






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


12 c

TV (g , [a; b]) = TV c (f , [a; b]) .


TV c (f , [0; t]) =: TV c (f , t) .


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.


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.


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.


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 .


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.


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.


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.


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.


Graphical interpretation of the Skorohod problem


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 .


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.


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.


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) .


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 .


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 .


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 .


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.


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.


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 .


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)}.


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.


Thank you!


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