Part II: Topics in fractional Brownian motion2. We have shown that fractional Brownian motion is not...

Part II: Topics in fractional Brownian motion

Esko Valkeila

Department of Mathematics and Systems Analysis

Spring School, Jena 26.3. 2011

We plan to discuss the following items during these lectures:

Fractional Brownian motion and its properties.

Representation results for fractional Brownian Motions.

Characterization theorems for fBm.

On stochastic integrals with respect to fBm.

Change of variables formula.

Mixed Brownian - fractional Brownian models.

Arbitrage.

Part II: Topics in fractional Brownian motion 2/47Esko Valkeila Spring School, Jena 26.3. 2011Department of Mathematics andSystems Analysis

On stochastics integrals with respect to fBm

For the rest of the lectures, the Hust index H is assumed to beH > 12 .We have shown that fractional Brownian motion is not asemimartingale. We will have at least the following problems:

The methods of classical stochastic analysis are not directlyapplicable.The change of limit and integral must be carefully justified.

Fractional Brownian motion is a Gaussian process, so one canapply the isometry of the covariance kernel to define Wienerintegrals, or one can use the Malliavin approach to developstochastic intergation theory with respect to fractional Brownianmotion; and there are many other possibilities.



Recent surveys on the different integration theories are given byBiagini et.al. (2008), Mishura (2008), and Coutin (2007).

We will use later so-called forward integrals. These are pathwiseintegrals, and they are applied to the case when the integrator is asum of standard Brownian motion and fractional Brownian motion,for example. When integrating with respect to fractional Brownianmotion, a useful integral is based on an extension ofRiemann-Stieltjes integrals. It turns out that so-called fractionalBesov spaces are useful here. We recall with some definitions.


On stochastics integrals with respect to fBmFractional Besov spaces

Fix 0 < β < 1.(i) Let Wβ1 = W

β1 [0,T ] be the space of real-valued measurable

functions f : [0,T ]→ R such that

‖f‖1,β := sup0≤s


The Besov-spaces are closely related to the spaces of Höldercontinuous functions. More precisely, for any 0 < � < β ∧ (1 − β),

Cβ+� [0,T ] ⊂ Wβ1 [0,T ] ⊂ Cβ−� [0,T ] and Cβ+� [0,T ] ⊂ Wβ2 [0,T ].

where Cγ[0,T ] stands for Hölder continuous functions of order γ.



Recall that the trajectories of BH for a.s. ω ∈ Ω, any T > 0 and any0 < γ < H belong to Cγ[0,T ]. This follows from the Kolmogorovcontinuity theorem. We obtain that the trajectories of BH for a.s.ω ∈ Ω, any T > 0 and any 0 < β < H belong to Wβ1 [0,T ].Recall the left-sided Riemann-Liouville fractional integral operatorIβ+ of order β > 0:

(Iβ0+f)(s) =1

Γ(β)

∫ s0

f(u)(s − u)β−1du.


On stochastics integrals with respect to fBmFractional integrals etc.

The corresponding right-sided fractional integral operator Iβ− wasalready defined by

(Iβt−f)(s) =1

Γ(β)

∫ ts

f(u)(u − s)β−1du.


On stochastics integrals with respect to fBmFractional integrals etc.

Let f : [0,T ]→ R and 0 < β < 1. If f ∈ Iβ+(L1[0,T ])(resp.f ∈ Iβ−(L∞[0,T ]) then the Weyl fractional derivatives are defined by

(Dβ0+f)(x) =1

Γ(1 − β)

(f(x)xβ

+ β

∫ x0

f(x) − f(y)(x − y)β+1

dy)

1(0,T)(x),(resp. (DβT− f)(x) =

1Γ(1 − β)

(f(x)

(T − x)β + β∫ T

x

f(x) − f(y)(y − x)β+1

dy)

1(0,T)(x)).



The following proposition clarifies the construction of the stochasticintegrals. This approach is by Nualart and Răscanu (2002), andinitiated by M. Zähle (1998).

Proposition Let f ∈ Wβ2 [0,T ], g ∈ W1−β1 [0,T ]. Then for any

t ∈ (0,T ] the Lebesgue integral∫ t0 (D

β0+f)(x)(D

1−βt− gt−)(x)dx

exists, and we can define the generalized Lebesgue-Stieltjesintegral by ∫ t

0fdg :=

∫ t0

(Dβ0+f)(x)(D1−βt− gt−)(x)dx.



The next theorem can be used to study the continuity of theintegral.Theorem [Nualart and Răscanu] Let f ∈ Wβ2 [0,T ] andg ∈ W1−β1 [0,T ]. Then we have the estimation

|∫ t

0fdg| ≤ 1

Γ(β)‖f‖2,β‖g‖1,1−β. (1)

Corollary Let f , fn ∈ Wβ2 [0,T ], ‖fn − f‖2,β → 0 as n → ∞, and

g ∈ W1−β1 [0,T ]. Then ∫fndg →

∫fdg.


Convex functions

We will prove a change of variables formula for convex functions.Therefore, we recall some results on convex functions. First, recallthat every convex function f : R→ R has a left-derivative f ′− and aright-derivative f ′+.The next theorem gives information about the left-derivative f ′− andright-derivative f ′+.Theorem The functions f

′− and f

′+ are increasing, respectively left

and right-continuous and the set {x : f ′−(x) , f′+(x)} is at most

countable.


Convex functions

Proposition The second derivative f′′

of convex function f existsin the sense of distributions, and it is a positive Radon measure;conversely, for any Radon measure µ on R, there is a convexfunction f such that f

′′= µ and for any interval I and x ∈ int(I) we

have the equality

f′−(x) =

12

∫Isgn(x − a)µ(da) + αI, (2)

where αI is a constant and sgn x = 1 if x > 0 and −1 if x ≤ 0.


Convex functions

If the supp(µ) is compact, then one can globally state that

f′−(x) =

12

∫sgn(x − a)µ(da) (3)

up to a constant term.


Change of variables formula

Let St = exp{BHt } be a geometric fractional Brownian motion,H ∈ ( 12 , 1), t ∈ [0,T ] and f : R→ R be a convex function. Then thestochastic integral ∫ T

0f′−(St )StdB

Ht (4)

can be understood in the sense of the generalizedLebesgue-Stieltjes integral a.s. ω ∈ Ω.


f ′−(St) is integrableOn the proof

Recall that the second derivative of f is a Radon measure µ. Weprove the result in the case that K := supp(µ) is a compact set.We will show that for β ∈ (1 − H, 12 ) we have that

‖f ′−(St )St‖2,β < ∞ a.s.

Recall the definition of the fractional Besov norm:

‖f‖2,β :=∫ T

0

|f(s)|sβ

ds +∫ T

0

∫ s0

|f(u) − f(s)|(u − s)1+β

duds < ∞.



Obviously,∫ T0

|f ′−(St )||St |tβ

dt ≤ maxt∈[0,T ]

(St )|f′−( max

t∈[0,T ]St )|

∫ T0

1tβ

dt < ∞ a.s.

For the second term in the definition of fractional Besov norm wemust work harder.



By the triangular inequality,∫ T0

∫ t0

|f ′−(St )St − f′−(Ss)Ss |

(t − s)β+1dsdt ≤ I1 + I2,

where, I1 =

∫ T0

∫ t0

|f ′−(St )||St − Ss |(t − s)β+1

dsdt ,

I2 =∫ T

0

∫ t0

|Ss ||f′−(St ) − f

′−(Ss)|

(t − s)β+1dsdt .



Using the Hölder continuity property of geometric fractionalBrownian motion trajectories one can bound from above I1 as

|I1| ≤ |f′−( max

t∈[0,T ]St )|C(ω)

∫ T0

∫ t0

(t − s)H−δ(t − s)β+1

dsdt < ∞ a.s.,

where δ ∈ (0,H − β) and C is a almost surely finite random variablesuch that

|St − Ss | ≤ C(ω)|t − s|H−δ.



We use the representation (3) to show I2 is finite almost surely.

|I2| ≤ maxt∈[0,T ]

(St )∫ T

0

∫ t0

|f ′−(St ) − f′−(Ss)|

(t − s)β+1dsdt ≤ I2,1 + I2,2,

where, I2,1 = max

t∈[0,T ](St )

∫ T0

∫ t0

∫K

1{Ss


By Tonelli’s theorem we have

E

∫ T0

∫ t0

∫K

1{Ss

f ′−(St) is integrableEnd of the proof

This implies∫ T0

∫ t0

∫K

1{Ss

Comments

One can show that the process S satisfies the following stochasticdifferential equation:

St = 1 +∫ t

0SudBBu ;

using this we can write (4) as∫ T0

f′−(St )dSt .


Comments

In the same lines one can show the pathwise stochastic integral∫ T0 f

′+(St )StdB

Ht

is well-defined in the sense of the generalized Lebesgue-Stieltjesintegral a.s. Moreover, we have∫ T

0f′−(St )StdB

Ht =

∫ T0

f′+(St )StdB

Ht .


Change of variables formula

Next we consider the Itô formula, which is more interesting for us.Theorem Let St = exp{BHt } be a geometric fractional Brownianmotion with H ∈ ( 12 , 1), t ∈ [0,T ] and f : R→ R be a convexfunction. Then

f(ST ) = f(S0) +∫ T

0f′−(St )StdB

Ht ,

where the stochastic integral is understood in the sense ofgeneralized Lebesgue-Stieltjes integral.

We indicate the idea of the proof.


Change of variables formulaOn the proof

The claim is true for smooth functions, because of the zeroquadratic variation of fBm.

We want to show that the equation (3) is valid for convex f ,where f ′ is replaced with the left derivative f ′− and the integralis generalized Lebesgue - Stieltjes integral. This is possible.


An example

The above results are true also for fractional Brownian motion BH,when H > 12 . If f : R→ R is a convex function with left-derivativef ′−, then the integral ∫ t

0f ′−(B

Hu )dB

Hu

exists as a generalized Lebesgue-Stieltjes integral. Moreover, wehave the change of variables formula

f(BHt ) − f(0) =∫ t

0f ′−(B

Hu )dB

Hu .


Comments of chang of variables formula

Take f(x) = |x | we have the following version of Tanaka’s formula

|BHT | =∫ T

0sgn(BHu )dB

Hu . (5)

So no local time appears in the formula.


Approximation

We prove the approximation result for integrals under theassumption that the second derivative of the convex function f hasa finite support.Theorem Assume T = 1 and let f be a convex function whichpositive Borel measure µ corresponding to its second derivativewith finite support K , i.e. µ(K) < ∞. Let ti = in ; i = 0, 1, ..., n.Then,

n∑i=0

f′−(Sti−1)(Sti − Sti−1)

a.s−→∫ 1

0f′−(St )dSt .


Uniquness of the integral representations

Consider the following problem:

The process g is an adapted process such that∫ T

0 gudBHu is

defined as a Riemann-Stieltjes integral. We know that

(?)

∫ T0

gudBHu = 0P − a.s.

When one can conclude that g = 0 Leb ⊗ P − a.s.?


Uniquness of the integral representations

If g is a deterministic process, then this is true.

Let g is a simple predictable process,

g =m∑

k=1

γi1[ti−i ,ti),

where γi ∈ L2(Fti−1 ,P), t̄ = {ti : 0 = t0 < t1 < · · · < tn = T } is apartition of the interval, and assume that (?) holds. Then onecan show that γi = 0, for example using the representation offractional Brownian motion with respect to standard Brownianmotion.


Zero problem

The next is by G. Shevchenko. We construct a bounded adaptedprocess ut such that

∫ t0 usdB

Hs → 0 a.s. as t → 1− and u is a.s.

bounded.Put bn = n3, an = n−2, ∆n = n−α/ζ(α), where α ∈ (1, 2H),ζ(α) =

∑n≥1 n−α, and define tn =

∑n−1k=1 ∆k so that tn → 1−,

n → ∞.Further, let τn = min{t ≥ tn :

∣∣∣∣BHt − BHtn ∣∣∣∣ ≥ an ∧ tn+1 andϕ(t) =

∞∑n=0

bn(BHt − BHtn )1[tn ,τn)(t).


Zero problem

Observe that ϕ(t) ≤ bnan = n on [tn, tn+1).Denote vt =

∫ t0 ϕ(s)dB

Hs . Then

vtn =12

n−1∑k=0

bk (BHτk − BHtk )

2 ≥ 12

n−1∑k=0

bk a2k 1Ak =12

n−1∑k=0

1k

1Ak ,

where An = {supt∈[tn ,tn+1]∣∣∣∣BHt − BHtn ∣∣∣∣ ≥ an}.


Zero problem

The following fact (a small ball probability estimate) is known.Assume that {Xt , t ∈ [0, 1]} is a centered Gaussian process withX0 = 0, whose increments satisfy

(X(t + h) − X(t)

)2= σ2(h),

where σ is non-decreasing and convex on [0, 1]. Then

P

supt∈[0,1]

|Xt | < σ(x) ≤ 2e−0.17/x .


Zero problem

By the self-similarity property of fractional Brownian motion,

{BHt − BHtn , t ∈ [tn, tn+1]}d= {∆1/Hn BHt , t ∈ [0, 1]}.

Hence, thanks to the above small ball probability estimate (withσ(h) = hH), we can write

P (Ω \ An) = P sup

t∈[0,1]|BHt | < an∆

−1/Hn

≤ 2 exp{−0.17

(an∆

−1/Hn

)−1/H} = 2 exp{−0.17 (n2n−α/H)1/H}= 2 exp{−0.17n(2−α/H)/H}.


Zero problem

Thus it is clear that∑

n≥1 P (Ω \ An) < ∞, so by the Borel–Cantellilemma for almost all ω there exists N(ω) so that ω ∈ An forn ≥ N(ω).Hence two conclusions.First, limt→1− vt = ∞ a.s. Indeed, vtn → ∞, n → ∞, but vt ≥ vtn fort ≥ tn, whence the assertion follows. Moreover, almost surely

vtn ≥∑

k :N(ω)≤k

Zero problem

Second, the functionut = ϕ(t)f ′(vt ),

where f(x) = xe−x2, is almost surely bounded. Indeed, it is clear

that |f ′(x)| ≤ Ce−x . Therefore, for t ∈ [tn, tn+1)

|ut | ≤ Cne−vt ≤ CneK(ω)−log n = C(ω).

Finally, by the Itô formula,∫ t0

usdBHs =∫ t

0ϕ(s)f ′(vs)dBHs =

∫ t0

f ′(vs)dvs = f(vt )→ 0, t → 1.


Transaction costs

We have seen that for a convex function f we have the following:

f(ST ) = f(S0) +∫ T

0f ′−(Su)dSu,

where S the geometric fractional Brownian motion

St = S0eBHt ; t ∈ [0, 1], and H > 1

2.

The two-assets market model consists of :

(i) Riskless asset (bond), Bt = 1; t ∈ [0, 1] which correspondsto zero interest rate.

(ii) Risky asset (stock) whose price is modeled by geometricfractional Brownian motion


Transaction costs

Now consider the discretized version of hedging strategy, i.e.

θnt =n∑

i=1

f′−(Stni−1)1(tni−1,tni ](t); t ∈ (0, 1].

In the presence of proportional transaction costs, the value of thisportfolio at terminal date is

V1(θn) = f(S0) +∫ 1

0θnt dSt − k

n∑i=1

Stni−1 |f′−(Stni ) − f

′−(Stni−1)|. (6)


Transaction costsLocal time of fBm

The occupation measure related to fractional Brownian motion isdefined by

ΓBH (I × U) = λ{t ∈ I : BHt ∈ U} =∫

I1{BHt ∈U}dt

where I and U are Borel sets on time interval [0,T ] and the realline respectively and λ stands for Lebesgue measure. It iswell-known that the occupation measure has a jointly continuousdensity (local time) which is denoted by lH(x, t) := lH(x, [0, t ]) andis Hölder continuous in t of any order α < 1 − H and in x of anyorder β < 1−H2H .


Transaction costs

The following result is by Azmoodeh (2010):

Theorem[Azmoodeh] Assume the level of transaction costsk = kn = k0n−(1−H), where k0 > 0. Then

P- limn→∞

V1(θn) = f(S1) − J,

where

J = J(k0) :=

√2π

k0

∫R

∫ 10

St lH(ln a, dt)µ(da),

and the inner integral in the right hand side is understood as limitof Riemann-Stieltjes sums a.s.


Transaction costsOn the proof

We have the indentity

V1(θn) − f(S1) = I1n − k0I2n,

where I1n =

n∑i=1

f′−(Stni−1)(Stni − Stni−1) −

∫ 10

f′−(St )dSt ,

I2n = ∆1−Hn

n∑i=1

Stni−1 |f′−(Stni ) − f

′−(Stni−1)|.



We already know that

I1n =n∑

i=1

f′−(Stni−1)(Stni − Stni−1) −

∫ 10

f′−(St )dSt → 0.

The study limiting behavious of the term I2n is done as follows. Weknow that we have

f ′−(x) =12

∫sgn(x − a)µ(da).



Assume that supp µ = {a}. One can then use the fact thatlocal time lH can be approximated with the level crossings ofpolygonal approximations to show that I2n → 0.

The next step is to assume that supp µ = {a1, . . . , ap}.The last step is based on the fact that on can approximate aconvex function using convex linear approximations [see thenext slide].



Assume that supp µ = {a}. One can then use the fact thatlocal time lH can be approximated with the level crossings ofpolygonal approximations to show that I2n → 0.The next step is to assume that supp µ = {a1, . . . , ap}.

The last step is based on the fact that on can approximate aconvex function using convex linear approximations [see thenext slide].



Assume that supp µ = {a}. One can then use the fact thatlocal time lH can be approximated with the level crossings ofpolygonal approximations to show that I2n → 0.The next step is to assume that supp µ = {a1, . . . , ap}.The last step is based on the fact that on can approximate aconvex function using convex linear approximations [see thenext slide].



Let f : R→ R be a convex function. For each interval [a, b], letπ = {a = a0 < a1 < ... < an = b} be a partition of the interval and

‖π‖ := max1≤i≤n

(ai − ai−1).

A piecewise linear function through points (ai , f(ai)) is called aconvex linear approximation of convex function f on the interval[a, b] based on the partition π. Let [a, b] be a closed interval and

{πm} be a sequence of partitions of the interval [a, b] such that

‖πm‖ → 0 as m → ∞

where πm = {a1, a2, ..., an(m)}.


On the proofApproximation

Let Pm be a convex linear approximation of convex function f onthe interval [a, b] based on partitions πm. Then we have(i) On the interval (a, b) as m → ∞

Pm → f and (Pm)′− → f

′− pointwise.

(ii) For any bounded continuous function g we have∫[a,b]

gdµm →∫[a,b]

gdµ as m → ∞,

where µm stands for Radon measure corresponding to thesecond derivative of Pm.


Transaction costs, local time and quadratic varia-tion

Let f be a convex function with positive Radon measure µ as itssecond derivative. Consider continuous semimartingaleXt = X0eWt with local time lX and X0 ∈ R+. By Itô-Tanaka formulawe have

f(X1) = f(X0) +∫ 1

0f′−(Xt )dXt +

12

∫R

lX (a, [0, 1])µ(da)

= f(X0) +∫ 1

0f′−(Xt )dXt +

12

∫R

alW (ln a, [0, 1])µ(da).

We have seen that the same structure appears in the limit withgeometric fractional Brownian motion with proportional transactioncosts.


Part II: Topics in fractional Brownian motion2. We have shown that fractional Brownian motion is not...

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