Riemann-Stieltjes integrals and fractional

Brownian motion

Esko Valkeila

Aalto University, School of Science and Engineering, Department of Mathematics

and Systems Analysis

Workshop on Ambit processes, Sandbjerg, Monday January 25,2010, 15.00–15.30

Co-authors and contents

The talk is based on joint work with Ehsan Azmoodeh (Aaltouniversity), Yuliya Mishura (Kiev) and Heikki Tikanmaki (Aaltouniversity).Azmoodeh and Tikanmaki are grateful to FGSS for support,Mishura acknowledges the support from SuomalainenTiedeakatemia, and V. acknowledges partial support from theAcademy of Finland.

Motivation

Co-authors and contents

The talk is based on joint work with Ehsan Azmoodeh (Aaltouniversity), Yuliya Mishura (Kiev) and Heikki Tikanmaki (Aaltouniversity).Azmoodeh and Tikanmaki are grateful to FGSS for support,Mishura acknowledges the support from SuomalainenTiedeakatemia, and V. acknowledges partial support from theAcademy of Finland.

Motivation

Stochastic integrals

Co-authors and contents

The talk is based on joint work with Ehsan Azmoodeh (Aaltouniversity), Yuliya Mishura (Kiev) and Heikki Tikanmaki (Aaltouniversity).Azmoodeh and Tikanmaki are grateful to FGSS for support,Mishura acknowledges the support from SuomalainenTiedeakatemia, and V. acknowledges partial support from theAcademy of Finland.

Motivation

Stochastic integrals

Main results

Co-authors and contents

The talk is based on joint work with Ehsan Azmoodeh (Aaltouniversity), Yuliya Mishura (Kiev) and Heikki Tikanmaki (Aaltouniversity).Azmoodeh and Tikanmaki are grateful to FGSS for support,Mishura acknowledges the support from SuomalainenTiedeakatemia, and V. acknowledges partial support from theAcademy of Finland.

Motivation

Stochastic integrals

Main results

Speculations

References

For fractional Brownian motion we use the book by Mishura:Stochastic Calculus for Fractional Brownian Motion and

Related Processes, 2008.

References

For fractional Brownian motion we use the book by Mishura:Stochastic Calculus for Fractional Brownian Motion and

Related Processes, 2008.

For the section Main results we use the paper Azmoodeh,Mishura, V.: On hedging European options in geometric

fractional Brownian motion market model, Statistics &Decisions (2010), forthcoming, and the manuscriptAzmoodeh, Tikanmaki, V. (2009), where we try to study thehedging of look-back options in geometric fractional Brownianmarket model.

References

For fractional Brownian motion we use the book by Mishura:Stochastic Calculus for Fractional Brownian Motion and

Related Processes, 2008.

For the section Main results we use the paper Azmoodeh,Mishura, V.: On hedging European options in geometric

fractional Brownian motion market model, Statistics &Decisions (2010), forthcoming, and the manuscriptAzmoodeh, Tikanmaki, V. (2009), where we try to study thehedging of look-back options in geometric fractional Brownianmarket model.

For the section Speculations we use the paper Bender,Sottinen, V. Pricing by hedging beyond semimartingales,Finance & Stochastics, 2008..

MotivationRepresentation theorem for Brownian functionals

We work with a probability space (Ω,F ,P).Let W be a standard Brownian motion, and F

W is the intrinsicfiltration of W : FW

t = σWs : s ≤ t.

MotivationRepresentation theorem for Brownian functionals

We work with a probability space (Ω,F ,P).Let W be a standard Brownian motion, and F

W is the intrinsicfiltration of W : FW

t = σWs : s ≤ t.The following representation theorem for square integrablefunctionals Y ∈ L2(FW

T ) is well-known:

F = EF +

∫ T

0

HFs dWs ;

here HF is a predictable process with the property

E∫ T

0

(

HFs

)2ds < ∞.

MotivationRepresentation theorem for Brownian functionals

We work with a probability space (Ω,F ,P).Let W be a standard Brownian motion, and F

W is the intrinsicfiltration of W : FW

t = σWs : s ≤ t.The following representation theorem for square integrablefunctionals Y ∈ L2(FW

T ) is well-known:

F = EF +

∫ T

0

HFs dWs ;

here HF is a predictable process with the property

E∫ T

0

(

HFs

)2ds < ∞.

Let BH be a fractional Brownian motion: BH is a continuouscentered Gaussian process with covariance

E(

BHt BH

s

)

=1

2

(

t2H + s2H − |t − s|2H)

.

MotivationRepresentation theorem for fractional Brownian functionals?

One can show that FWt = FBH

t , 0 ≤ t ≤ T . Hence, if the randomvariable F ∈ L2(FW

T ), then we automatically have that

F ∈ L2(FBH

T ). In plain English: every Brownian functional is also afractional Brownian functional.

MotivationRepresentation theorem for fractional Brownian functionals?

One can show that FWt = FBH

t , 0 ≤ t ≤ T . Hence, if the randomvariable F ∈ L2(FW

T ), then we automatically have that

F ∈ L2(FBH

T ). In plain English: every Brownian functional is also afractional Brownian functional.It is then natural to ask, if every square integrable fractionalBrownian functional has a representation as a stochastic integralwith respect to fractional Brownian motion?

MotivationRepresentation theorem for fractional Brownian functionals?

One can show that FWt = FBH

t , 0 ≤ t ≤ T . Hence, if the randomvariable F ∈ L2(FW

T ), then we automatically have that

F ∈ L2(FBH

T ). In plain English: every Brownian functional is also afractional Brownian functional.It is then natural to ask, if every square integrable fractionalBrownian functional has a representation as a stochastic integralwith respect to fractional Brownian motion?Now we will have some problems:

Fractional Brownian motion is not a semimartingale, and thenthe definition of the integrals is not clear at all.

In contrast to the Brownian case, we already have the firstnegative result: if Y ∈ spanBH

s : s ≤ t for H > 12, then Y

may fail to have a representation as Y =∫ t

0fsdBH

s , where f isa deterministic function [Pipiras & Taqqu, Molchan].

Stochastic intgeralsRepresentation theorem with the help of abstract ’integrals’

Skorohod integrals and Malliavin calculus are tools to exportresults from Wiener space to other Gaussian spaces. One canfollow this approach and prove the following type of result: ifF ∈ L2(FBH

T ), then one has the following representation for F

F = EF +

∫ T

0

HFs δBH

s ,

and this ’integral’ is understood as Skorohod ’integral’, or as anadjoint of Malliavin derivative.

Stochastic intgeralsRepresentation theorem with the help of abstract ’integrals’

Skorohod integrals and Malliavin calculus are tools to exportresults from Wiener space to other Gaussian spaces. One canfollow this approach and prove the following type of result: ifF ∈ L2(FBH

T ), then one has the following representation for F

F = EF +

∫ T

0

HFs δBH

s ,

and this ’integral’ is understood as Skorohod ’integral’, or as anadjoint of Malliavin derivative.But we continue to have problems:

If you want that for s < t the following holds

∫ t

0

HuδBHu =

∫ s

0

HuδBHu +

∫ t

s

HuδBHu ,

one must use non-anticipative integrands H.

Stochastic integralsRepresentation theorem with the help of abstract ’integrals’; Stieltjes integrals

The integration theory with abstract ’integrals’ is difficult tointerpret in some applications. On the other hand, Stieltjesintegrals with respect to fractional Brownian motion, H > 1

2, can

be reasonably interpreted in these applications.

Stochastic integralsRepresentation theorem with the help of abstract ’integrals’; Stieltjes integrals

The integration theory with abstract ’integrals’ is difficult tointerpret in some applications. On the other hand, Stieltjesintegrals with respect to fractional Brownian motion, H > 1

2, can

be reasonably interpreted in these applications.We have now different type of problems:

Which random variables Y ∈ L2(FBH

T ) have a representation

Y = C +

∫ T

0

HYs dBH

s ,

where the integral is a Riemann-Stieltjes integral.

Fact: if Y = F (BHT ) with F ∈ C1(R), then

Y = F (0) +

∫ T

0

Fx(BHs )dBH

s .

Main resultsConvex functions of B

HT

The next result is by Azmoodeh, Mishura, V.: Assume that F is aconvex function with the right derivative F+

x . Then we have therepresentation

F (BHT ) = F (0) +

∫ T

0

F+x (BH

s )dBHs . (1)

What is a bit surpising in (1) is the fact that the integral onthe right hand side is a Riemann-Stieltjes integral: if oneapplies the change of variables formula (1) to the functionf (x) = |x | one obtains

|BHT | =

∫ T

0

sgn(BHs )dBH

s ,

and the process sgn(BH) has unbounded variation oncompacts.

Main resultsRunning maximum of a continuous bounded variation function A

The change of variables formula (1) is the same for a continuousbounded variation functions A: if F is convex, then

F (AT ) = F (A0) +

∫ T

0

F+x (As)dAs .

A natural question: to what extend fractional Brownianmotion behaves as a continuous function with boundedvariation?

Main resultsRunning maximum of a continuous bounded variation function A

The change of variables formula (1) is the same for a continuousbounded variation functions A: if F is convex, then

F (AT ) = F (A0) +

∫ T

0

F+x (As)dAs .

A natural question: to what extend fractional Brownianmotion behaves as a continuous function with boundedvariation?

Representation of the running maximum A∗t = maxs≤t As of a

continuous bounded variation function A with A0 = 0:

A∗t =

∫ t

0

1A∗

s =AsdAs ; (2)

this is a result by Azmoodeh, Tikanmaki, V. [work in progress].

Main resultsRunning maximum of a continuous bounded variation function A

The change of variables formula (1) is the same for a continuousbounded variation functions A: if F is convex, then

F (AT ) = F (A0) +

∫ T

0

F+x (As)dAs .

A natural question: to what extend fractional Brownianmotion behaves as a continuous function with boundedvariation?

Representation of the running maximum A∗t = maxs≤t As of a

continuous bounded variation function A with A0 = 0:

A∗t =

∫ t

0

1A∗

s =AsdAs ; (2)

this is a result by Azmoodeh, Tikanmaki, V. [work in progress].

Apparently (2) does not generalize to fractional Brownianmotion.

Speculations

Consider the process X ǫ, where X ǫ

t = BHt + ǫWt ; here the

Brownian motion W is independent of the fractional Brownianmotion of BH , H > 1

2.

Assume now that the random variableF = F (WT , η1, η2, . . . , ηk) ∈ FW

T has an integral representation

F = c +

∫ T

0

f (Ws , η1s , . . . , η

ks )dWs ,

where ηj are so-called hindsight functionals like W ∗s ,

∫ s

0Wudu etc.

Speculations

Consider the process X ǫ, where X ǫ

t = BHt + ǫWt ; here the

Brownian motion W is independent of the fractional Brownianmotion of BH , H > 1

2.

Assume now that the random variableF = F (WT , η1, η2, . . . , ηk) ∈ FW

T has an integral representation

F = c +

∫ T

0

f (Ws , η1s , . . . , η

ks )dWs ,

where ηj are so-called hindsight functionals like W ∗s ,

∫ s

0Wudu etc.

It follows from recent work by Bender, Sottinen, V. that thefunctional F ǫ with respect to the paths of X ǫ has the same integralrepresentation :

F ǫ = c +

∫ T

0

f (X ǫ

s , η1s (ǫ), . . . , η

ks (ǫ))dX ǫ

s .

Speculations and closing

If one now lets ǫ → 0 in

F ǫ = c +

∫ T

0

f (X ǫ

s , η1s (ǫ), . . . , η

ks (ǫ))dX ǫ

s .

,when it is true that one gets

FH = c +

∫ T

0

f (BHs , η1

s (H), . . . , ηks (H))dBH

s ?

Speculations and closing

If one now lets ǫ → 0 in

F ǫ = c +

∫ T

0

f (X ǫ

s , η1s (ǫ), . . . , η

ks (ǫ))dX ǫ

s .

,when it is true that one gets

FH = c +

∫ T

0

f (BHs , η1

s (H), . . . , ηks (H))dBH

s ?

I hope that before my next talk on this topic we will have somenew results on this.

Speculations and closing

If one now lets ǫ → 0 in

F ǫ = c +

∫ T

0

f (X ǫ

s , η1s (ǫ), . . . , η

ks (ǫ))dX ǫ

s .

,when it is true that one gets

FH = c +

∫ T

0

f (BHs , η1

s (H), . . . , ηks (H))dBH

s ?

I hope that before my next talk on this topic we will have somenew results on this.

Thank you!