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Potential Analysis 10: 177–214, 1999. © 1999 Kluwer Academic Publishers. Printed in the Netherlands. 177 Stochastic Analysis of the Fractional Brownian Motion L. DECREUSEFOND and A.S. ÜSTÜNEL École Nationale Supérieure des Télécommunications, 46, Rue Barrault, F-75634 Paris, France. (e-mail: [email protected] and [email protected]) (Received: 21 February 1997; accepted: 21 February 1997) Abstract. Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations. Mathematics Subject Classifications (1991): Primary: 60H07; Secondary: 60G18. Key words: Fractional Brownian motion, stochastic calculus of variations, Itô formula, Girsanov formula. 1. Introduction In engineering applications of probability, stochastic processes are often used to model the input of a system. For instance, the financial mathematics requires stochastic models for the time evolution of assets and the queuing networks analysis is based on models of the offered traffic. Hitherto, the stochastic processes used in these fields are often supposed to be Markovian. However, recent studies [8] show that real inputs exhibit long-range dependence: the behavior of a real process after a given time t does not only depend on the situation at t but also of the whole history of the process up to time t . Moreover, it turns out that this property is far from being negligible because of the effects it induces on the expected behavior of the global system [12]. Another property that have the processes encountered in applications (at least in communication networks) is the self-similarity (see [8]): their behavior is stochastic- ally the same, up to a space-scaling, whatever the time-scale is - this is to say that the process {X αt ,t ∈[0, 1]} has the same law as the process {α H X t ,t [0, 1]}, where H is called the Hurst parameter. Several estimations on real data tend to show that H often lies between 0.7 and 0.8 whereas for instance, the usual Brownian motion has a Hurst parameter equal to 0.5 but it is also clear that some real processes have a Hurst parameter less than 0.5 – see [5]. There exist several stochastic processes which are self-similar and exhibiting long-range dependence but the fractional Brownian motion (fBm for short) seems to be one of the simplest. 135625.tex; 15/08/1995; 7:44; p.1 corrected JEFF INTERPRINT pota411 (potakap:mathfam) v.1.15

Stochastic Analysis of the Fractional Brownian Motion

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Page 1: Stochastic Analysis of the Fractional Brownian Motion

Potential Analysis10: 177–214, 1999.© 1999Kluwer Academic Publishers. Printed in the Netherlands.

177

Stochastic Analysis of the FractionalBrownian Motion

L. DECREUSEFOND and A.S. ÜSTÜNELÉcole Nationale Supérieure des Télécommunications, 46, Rue Barrault, F-75634 Paris, France.(e-mail: [email protected] and [email protected])

(Received: 21 February 1997; accepted: 21 February 1997)

Abstract. Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculuscannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula,the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractionalBrownian motion using the stochastic calculus of variations.

Mathematics Subject Classifications (1991):Primary: 60H07; Secondary: 60G18.

Key words: Fractional Brownian motion, stochastic calculus of variations, Itô formula, Girsanovformula.

1. Introduction

In engineering applications of probability, stochastic processes are often used tomodel theinputof a system. For instance, the financial mathematics requires stochasticmodels for the time evolution of assets and the queuing networks analysis is basedon models of the offered traffic. Hitherto, the stochastic processes used in thesefields are often supposed to be Markovian. However, recent studies [8] show thatreal inputsexhibit long-range dependence: the behavior of a real process after agiven timet does not only depend on the situation att but also of the whole historyof the process up to timet . Moreover, it turns out that this property is far frombeing negligible because of the effects it induces on the expected behavior of theglobal system [12].

Another property that have the processes encountered in applications (at least incommunication networks) is the self-similarity (see [8]): their behavior is stochastic-ally the same, up to a space-scaling, whatever the time-scale is - this is to saythat the process{Xαt, t ∈ [0,1]} has the same law as the process{αHXt, t ∈[0,1]}, whereH is called the Hurst parameter. Several estimations on real datatend to show thatH often lies between 0.7 and 0.8 whereas for instance, the usualBrownian motion has a Hurst parameter equal to 0.5 but it is also clear that somereal processes have a Hurst parameter less than 0.5 – see [5]. There exist severalstochastic processes which are self-similar and exhibiting long-range dependencebut the fractional Brownian motion (fBm for short) seems to be one of the simplest.

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178 L. DECREUSEFOND AND A.S. ÜSTÜNEL

DEFINITION 1.1. For anyH in (0, 1), the fractional Brownian motion of index(Hurst parameter)H, {WH

t ; t ∈ [0,1]} is the centered Gaussian process whosecovariance kernel is given by

RH(s, t) = EH [WHs W

Ht ] def= VH

2(s2H + t2H − |t − s|2H ),

where

VHdef= 0(2− 2H) cos(πH)

πH(1− 2H).

Since forH 6= 1/2, the fBm is not a semimartingale, we can not use the usualstochastic calculus to analyze it, however since it is a Gaussian process, we canapply the stochastic calculus of variations which is valid on general Wiener spaces.Actually, two choices are offered to us: either some well known properties of thestandard Brownian motion are used to derive some properties of the fBm or we canproceed by an intrinsic analysis of the fBm. The first approach leads us to the Itô–Clark formula whereas the Itô formula and the Girsanov theorem are more intrinsicresults. This paper is organized as follows: in Section 2, we give some results onhypergeometric functions and deterministic fractional calculus which will be usefulin the sequel, in Section 3 we give some sample-paths properties of the fractionalBrownian motion, in Section 4 we introduce the stochastic calculus of variations.It enables us to define several stochastic integrals with respect to the fractionalBrownian motion of any order. We can then give the Itô–Clark representationformula and the Girsanov theorem for adopted processes. In the last section, wegive Itô formulae forH > 1/2 using different stochastic integrals. Throughout thepaper, we give two practical applications such as the simulation of sample-paths ofthe fractional Brownian motion and an estimation problem involving an fBm.

2. Deterministic Fractional Calculus

The Gauss hypergeometric functionF(a, b, c, z) (for details, see [11]) is definedfor anya, b, anyz, |z| < 1 and anyc 6= 0,−1, . . . by

F(a, b, c, z)def=+∞∑k=0

(a)k(b)k

(c)kk! zk, (1)

where(a)0 = 1 and(a)kdef= 0(a + k)/0(a) = a(a + 1) . . . (a + k − 1) is the

Pochhammer symbol. Ifa or b is a negative integer the series terminates aftera finite number of terms andF(a, b, c, z) is a polynomial inz. The radius ofconvergence of this series is 1 and there exists a finite limit whenz tends to 1

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 179

(z < 1) provided thatR(c− a− b) > 0. For anyz such that|arg(1− z)| < π , anya, b, c such thatR(c) > R(b) > 0, F can be defined by

F(a, b, c, z)def= 0(c)

0(b)0(c − b)∫ 1

0ub−1(1− u)c−b−1(1− zu)−a du. (2)

Given (a, b, c), consider6 the set of triples(a′, b′, c′) such that|a − a′| = 1 or|b − b′| = 1 or |c − c′| = 1. Any hypergeometric functionF(a′, b′, c′, z) with(a′, b′, c′) in 6 is said to be contiguous toF(a, b, c). For any two hypergeometricfunctionsF1 andF2 contiguous toF(a, b, c, z), there exists a relation of the type:

P0(z)F (a, b, c, z)+ P1(z)F1(z)+ P2(z)F2(z) = 0, for z, |arg(1− z| < π, (3)

where for anyi, Pi is a polynomial with respect toz. These relations permit todefine the analytic continuation ofF(a, b, c, z) with respect to its four variables inthe domainC× C× (C\{0.− 1.− 2, . . .})× {z, |arg(1− z)| < π}. We will alsouse other types of relations between different hypergeometric functions, namely:

F(a, b, c, z) = 0(c)0(b − a)0(c − a)0(b)(1− z)

−aF (a, c − b,1+ a − b,1/(1− z))

+0(c)0(a − b)0(c − b)0(a)(1− z)

−bF (b, c − a,1− a + b,1/(1− z)), (4)

for anyz such that|arg(1− z)| < π anda − b 6= 0,±1,±2, . . . We now considersome basic aspects of the deterministic fractional calculus− the main referencefor this subject is [13].

DEFINITION 2.1. Letf ∈ L1([a, b]), the integrals

(I αa+f )(x)def= 1

0(α)

∫ x

a

f (t)(x − t)α−1 dt, x > a,

(I αb−f )(x)def= 1

0(α)

∫ b

x

f (t)(x − t)α−1 dt, x 6 b,

whereα > 0, are respectively called right and left fractional integral of the orderα.

For anyα > 0, anyf ∈ Lp([0,1]) andg ∈ Lq([0,1]) wherep−1 + q−1 6 α,we have:∫ t

0f (s)(I α0+g)(s)ds =

∫ t

0(I αt−f )(s)g(s)ds. (5)

DEFINITION 2.2. Forf given in the interval[a, b], each of the expressions

(Dαa+f )(x)

def=(

d

dx

)[α]+1

I1−{α}a+ f (x),

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180 L. DECREUSEFOND AND A.S. ÜSTÜNEL

(Dαb−f )(x)

def=(− d

dx

)[α]+1

I1−{α}b− f (x),

are respectively called the right and left fractional derivative (proved they exist),where[α] denotes the integer part ofα and{α} = α − [α].

A sufficient condition forf to be α-differentiable almost everywhere (withrespect to the Lebesgue measure on[a, b]) is thatf is continuously differentiableof any integer order less than[α] and thatf ([α]) is absolutely continuous. Notethat(D1

a+f ) coincides with the usual derivative of absolutely continuous function.Moreover, iff is α-differentiable thenf is β-differentiable for anyβ 6 α.

PROPOSITION 2.1.For α ∈ C such thatR(α) > 0, we have:

Dαa+I

αa+f = f for f ∈ L1([a, b]),

I αa+Dαa+f = f for f ∈ I αa+(L1([a, b])).

As a consequence, we will often denoteDαa+ by I−α

a+ . Moreover, forp > 1, thelatter proposition also induces that a functionf in I α

a+(Lp([a, b])) is α-different-

iable (for a reciprocal of this assertion, see [13, p. 232]) and hence continuous.Some extra work proves that such a function is Hölder continuous of orderα −1/p-[13, Theorem 3.6, p. 67]. The next theorem will be a key result for the sequel:

THEOREM 2.1 (cf [13, p. 187]).For H ∈ (0,1), consider the integral transform:

(KHf )(t) = 0(H + 1/2)−1∫ t

0(t − x)H−1/2

×F(H − 1/2,1/2−H,H + 1/2,1− t/x)f (x)dx. (6)

KH is an isomorphism fromL2([0,1]) ontoIH+1/20+ (L2([0,1])) and

KHf = I 2H0+ x

1/2−H0+ I

1/2−H0+ xH−1/2f forH 6 1/2,

KHf = I 10+x

H−1/2IH−1/20+ x1/2−Hf forH > 1/2.

Note that ifH > 1/2, r → KH(t, r) is continuous on(0, t] so that we can includet in the indicator function.

3. Properties of the Fractional Brownian Motion

Using the Kolmogorov criterion, it is easy to see that for anyH , there exists aversion ofWH whose sample-paths are continuous and with standard techniques, it

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 181

Figure 1. Typical sample-path forH = 0.2,H = 0.5,H = 0.8.

can also be shown that sample-paths are nowhere differentiable (see [10]). Further-more, the form of the covariance kernel entails thatWH has stationary incrementsand that the process is self-similar in the sense that

{WHαt , t ∈ [0,1]} d= {αHWH

t , t > 0}.Note that increments are independent only whenH = 1/2, forH > 1/2, incre-ments are positively correlated and forH < 1/2 they are negatively correlated.This difference of behavior between the casesH < 1/2 andH > 1/2 can also beseen in the regularity of sample-paths as show the next figure and the next theorem.

THEOREM 3.1. LetH ∈ (0,1), the sample-paths ofWH are a.s. Hölder continu-ous only of order less thanH .

Proof.Since for anyα > 0, we have

EH [|WHt −WH

s |α] = Cα|t − s|Hα,the Kolmogorov criterion implies that the sample-paths ofWH are almost surelyHölder continuous of any order less thanH .

As a consequence of the results in [1], we have

PH

(lim supu→0+

WHu

uH√

log log u−1= √VH) = 1.

Hence it is impossible forWH to have sample-paths Hölder continuous of an ordergreater thanH . 2LetW = C0([0,1],R) be the Banach space of continuous functions, null at time0, equipped with the sup-norm andW ∗ be its topological dual. For anyH ∈(0,1),PH is the unique probability measure onW such that the canonical process(Ws; s ∈ [0,1]) is a centered Gaussian process with covariance kernelRH :

EH [WsWt ] = RH(s, t).The canonical filtration is given byF H

t = σ {Ws, s 6 t} ∨ NH andNH is theset of thePH -negligible events. LetHH be the Cameron–Martin space associated

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182 L. DECREUSEFOND AND A.S. ÜSTÜNEL

with (W,PH ): the unique Hilbert space (identified with its dual) continuously anddensely embedded inW such that, for anyη in W ∗,∫

W

ei〈η,w〉w∗,w dPH(w) = exp(−‖η‖2HH

/2), (7)

whereη is the image ofη under the injectionW ∗ ⊂ HH . In order to be able todescribeHH , we need the following preliminary lemma:

LEMMA 3.1. For anyH ∈ (0,1), RH (s, t) can be written as

RH(s, t) =∫ 1

0KH(s, r)KH (t, r)dr, (8)

in operator notations,RH = KHK∗H , whereKH is the Hilbert–Schmidt operator

introduced in Theorem[2.1]. We hereafter identify an operator and its kernel.Proof.ForH > 1/2, it is easy to see that

RH(s, t) = VH

4H(2H − 1)

∫ t

0

∫ s

0|r − u|2H−2 dudr

Moreover (see [2]),

VH

4H(2H − 1)|r − u|2H−2

= (ru)H−1/2∫ r∧u

0v1/2−H(r − v)H−3/2(u− v)H−3/2 dv.

Hence forH > 1/2, (8) holds with

KH(t, r) = r1/2−H

0(H − 1/2)

∫ t

r

uH−1/2(u− r)H−3/2 du1[0,t ](r).

A change of variable in this equation transforms the integral term in

(t − r)H−1/2rH−1/2∫ 1

0uH−3/2(1− (1− t/r)u)H−1/2 du.

By the definition (2) of hypergeometric functions, we see that (6) holds true forH > 1/2. Using property (4), we have

KH(t, r) = 2−2H√π0(H) sin(πH)

rH−1/2

+ 1

20(H + 1/2)(t − r)H−1/2F

(1/2−H,1,2− 2H,

r

t

).

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 183

If H < 1/2 then the hypergeometric function of the latter equation is continuouswith respect tor on [0, t] because 2− 2H − 1− 1/2+H = 1/2−H is positive.Hence, forH < 1/2,KH (t, r)(t − r)1/2−Hr1/2−H is continuous with respect toron [0, t]. ForH > 1/2, the hypergeometric function is no more continuous int butwe have [11]:

F(1/2−H,1,2− 2H,

r

t

)= C1F(1/2−H,1,H + 1/2,1− r/t)

+C2(1− r/t)1/2−H(r/t)2H−1.

Hence, forH > 1/2,KH (t, r)rH−1/2 is continuous with respect tor on [0, t]. Fixδ ∈ [0,1/2) andt ∈ (0,1], we have:

|KH(t, r)| 6 Cr−|H−1/2|(t − r)−(1/2−H)+1[0,t ](r)

whereC is uniform with respect toH ∈ [1/2−δ,1/2+δ]. Thus, the two functionsdefined on{H ∈ C, |H − 1/2| < 1/2} by

H ∈ (0,1) 7→ RH(s, t)andH ∈ (0,1) 7→∫ 1

0KH(s, r)KH (t, r)dr

are well defined, analytic with respect toH and coincide on[1/2,1) thus they areequal for anyH ∈ (0,1) and anys andt in [0,1]. 2In the previous proof we proved a result which is so useful in its own that it deservesto be a theorem:

THEOREM 3.2. For anyH ∈ (0,1), there exist a constantcH such that for anytandr, we have:

|KH(t, r)| 6 cH r−|H−1/2|(t − r)−(1/2−H)+1[0,t ](r), (9)

wherex+ = max(x,0).

THEOREM 3.3.

1. HH = {KH h; h ∈ L2([0,1],dt)}, i.e., anyh ∈ HH can be represented as

h(t) = KH h(t) def=∫ 1

0KH(t, s)h(s)ds,

where h belongs toL2([0,1]). For any HH -valued random variableu, wehereafter denote byu theL2([0,1];R)-valued random variable such that

u(w, t) =∫ t

0KH(t, s)u(w, s)ds.

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184 L. DECREUSEFOND AND A.S. ÜSTÜNEL

2. The scalar product onHH is given by

(h, g)HH= (KH h,KH g)HH

def= (h, g)L2([0,1]).

3. The injectionRH fromW ∗ intoHH can be decomposed asRHη = KH(K∗Hη).Furthermore, the restriction ofK∗H toW ∗ is the injection fromW ∗ intoL2([0,1]):

W ∗K∗H- L2([0,1];R) KH- HH

iH- W.

REMARK 3.1. Note that as a vector space,HH is equal toIH+1/20+ (L2([0,1])) but

the norm on each of these spaces are different since the norm of an elementh inthe latter space is theL2 norm ofI−H−1/2

0+ (h).Proof.From Theorem [2.1], we know thatKH is bijection fromL2([0,1]) onto

IH+1/20+ (L2([0,1])) ⊂ W . For anyα > −1/2, (KHxα)(t) = cα,H tα+H+1/2, hence

HH contains all the polynomials null at 0 so thatHH is dense inW from Stone–Weierstrass theorem.

Let iH denote the inclusion fromHH into W, i∗H the inclusion fromW ∗ intoHH , jH the canonical identification isomorphism betweenH∗H andHH andRH =jH ◦ i∗H , i.e.,RH is theembeddingof W ∗ into HH . Forη ∈ W ∗, we have on onehand

(RH(η), h)HH

def=∫ 1

0

˙RH(η)(s)h(s)ds

and on the other hand,

(RH(η), h)HH= 〈η.iH (h)〉W∗W =

∫ 1

0

∫ 1

0KH(t, s)h(s)dsη(dt)

=∫ 1

0(K∗Hη)(s)h(s)ds,

whereK∗H is the adjoint ofKH for theL2([0,1]) scalar-product. It follows that˙

RH(η) = K∗Hη and then thatRH = KH ◦K∗H .It remains to prove (7); for we compute theHH norm ofRH(η):∫〈η,w〉2W∗,W dPH(w) = EH

[∫ 1

0

∫ 1

0WsWtη(ds)η(dt)

]

=∫ 1

0

∫ 1

0RH(t, s)η(ds)η(dt)

=∫ 1

0

∫ 1

0

∫ 1

0KH(t, r)KH (s, r)drη(ds)η(dt)

= ‖K∗Hη‖2L2([0,1]) = ‖RH(η)‖2HH.

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 185

HenceHH has the three properties defining the Caneron–Martin space of theWiener space(W,PH ). 2COROLLARY 3.1. The fractional Brownian motion has the representation in law:

WHt =

∫ t

0KH(t, s)dW 1/2

s .

Proof. The process(∫ t

0 KH(t, s)dW 1/2s , t ∈ [0,1]) is Gaussian with the con-

venient covariance kernel. 2REMARK 3.2. We will show in the sequel that this representation holds in thetrajectorial sense with a fixed, standard Brownian motion constructed on(W,PH ).The representation of the corollary is different from the one in [10] since it requiresonly one standard Brownian motion instead of two.

The computer simulation of fBm paths is a classical problem of numerical analysis,the difficulty being due to the nontrivial correlation between the increments. Thefollowing proposition gives an approximation scheme:

PROPOSITION 3.1.Letπn be an increasing sequence of partitions of[0,1] suchthat the mesh|πn| of πn tends to0 asn goes to infinity. The sequence of processes{Wn, n > 0} defined by

Wnt =

∑tni ∈πn

1

tni+1 − tni

∫ tni+1

tni

KH (t, s)ds(W 1/2tni+1−W 1/2

tni)

converges toWH in L2(P1/2⊗ ds).Proof. Let Gn be theσ -field generated by{W 1/2

tni, tni ∈ πn}. For t fixed, the

sequence

Wnt

def= E1/2

[∫ t

0KH(t, s)dW 1/2

s |Gn]

is a square integrable martingale with respect to the filtration(Gn). Moreover, usingthe Jensen inequality,

supn

E1/2

[∫ 1

0(Wn

t )2 dt

]6∫ 1

0RH(t, t)dt < +∞.

Hence the sequence(Wn)n is anL2([0,1])-valued(Gn) square integrable martin-gale so that it convergesPH almost everywhere and inL2(W ;L2([0,1])), i.e.,

limn→+∞ E1/2

[∫ 1

0(Wn

t −Wt)2 dt

]= 0.

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186 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Furthermore, a simple calculation (using the Gaussian character and independenceof the random variablesW 1/2

tni+1−W 1/2

tni) shows that

E1/2

[∫ t

0KH(t, s)dW 1/2

s |Gn]

=∑tni ∈πn

1

tni+1 − tni

∫ tni+1

tni

KH (t, s)ds(W 1/2tni+1−W 1/2

tni)

which proves the result. 2

4. Preliminaries and the Malliavin Calculus

For details on the construction of Malliavin calculus on Wiener spaces, we refer to[15, 17]. As for all Gaussian spaces, we have (see [4])

THEOREM 4.1 (Cameron–Martin Theorem).For anyRHη ∈ HH ,

EH [F(w + RHη)] =∫F(w) exp(〈η,w〉 − ‖RHη‖2HH

/2)dPH(w). (10)

DEFINITION 4.1. LetX be a separable Hilbert space andF :W → X andX-valued functional of the form

F(w) =k∑i=1

fi(〈l1, w〉, . . . , 〈ln, w〉)xi (11)

where for eachi ∈ {1, . . . , n}, li is in W ∗ andxi belongs toX. F is said to bea smooth cylindricfunctional (respectively apolynomial) whenfi is an elementof the Schwartz spaceS(Rn) (respectively of the set of real polynomials withn variables). We will denote byS(X) (respectivelyP (X)) the set ofX-valuedsmooth cylindric functionals and simplyS (respectivelyP ) whenX = R.

ConsiderCH0 = R and for n > 0, defineCHn as the closed vector spacespanned inL2(PH) by the elements ofP of degree less thann. SetCH

0 = CH0and supposeCH

1 , . . . ,CHn are defined, then, we defineCH

n+1 as the orthogonal com-plement ofCH

1 ⊕ · · · ⊕CHn in CHn+1. As for all Gaussian spaces, we have the chaos

decomposition:

THEOREM 4.2.

L2(PH) =⊕n>0

CHn .

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 187

This means that everyPH -square integrable functional fromW to R can bewritten in a unique way as

F =+∞∑n=0

JHn F, (12)

whereJHn is the orthogonal projection ofL2(PH) ontoCHn .

DEFINITION 4.2 (Ornstein–Uhlenbeck semi-group). ForS 3 F :W → R in∪p>1L

p(PH) andt > 0, we define(T Ht F )(w) by the Mehler formula:

T Ht F (w) =∫F(e−tw +

√1− e−2t y)dPH (y).

It turns out that forF ∈ S, (t 7→ T Ht F ) is differentiable inLp with respect tot , so that we can define the Ornstein–Uhlenbeck operatorLH :

DEFINITION 4.3 (Ornstein–Uhlenbeck operator).LH is defined onS by

LHF(w) = d

dtTtF (w)

∣∣∣∣t=0

.

The operatorLH can then be extended onLp(PH) as the infinitesimal gener-ator of a contraction semi-group onLp(PH); let Domp(LH) be the domain of theextension ofLH in Lp(PH).

DEFINITION 4.4 (Sobolev spaces). Letp > 1, q such thatp−1 + q−1 = 1 andk ∈ Z. DHp,k(X) is the completion ofS(X) with respect to the norm

‖F‖p,k,H def= ‖(I −LH)k/2F‖LpH ,

where

(I −LH)k/2F =

+∞∑n=0

(1+ n)k/2JHn F.

It is well known thatDHp,k(X) is the dual ofDHp,−k with the duality pairing

〈F,G〉 = EH [((I −L)k/2F, (I −L)−k/2G)X]and that for anyk′ 6 k,

DHp,k(X) ⊂ DHp,k′(X) ⊂ DHp,0 = Lp(W ;X) ⊂ DHp,−k′(X) ⊂ DHp,−k(X).

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188 L. DECREUSEFOND AND A.S. ÜSTÜNEL

We also introduce the spacesDHp,k,a(HH) which are for anyp > 1 and anyk ∈ Nthe subspaces ofDHp,k(HH) composed by the adapted processes. For anyk ∈−N, DHp,k,a(HH) is the dual ofDHp,−k,a(HH). The notationDH∞(X) (respectivelyDH∞,a(HH)) will stand for ∩p,k>0DHp,k(X) (respectively∩p,k>0DHp,k,a(HH)) andDH−∞(X) = ∪p,−k>0Dp,k(X), respectivelyDH−∞,a(HH) = ∪p,−k>0DHp,k,a(HH).

DEFINITION 4.5. ForF in S(X), theH -Gross–Sobolev derivative ofF , denotedby∇F and is theHH ⊗X-valued mapping defined by

∇F(w) =n∑i=1

∂f

∂xi(〈l1, w〉, . . . , 〈ln, w〉)RH(li)⊗ x. (13)

REMARK 4.1. Takeli = εti , the Dirac measure at timeti and let

F(w) = f (〈εt1, w〉, . . . , 〈εtn , w〉)= f (Wt1, . . . ,Wtn),

then we have by (13),

∇F(w) =n∑i=1

∂f

∂xi(Wt1, . . . ,Wtn)RH(εti ),

i.e.,

∇F(w)(s) =n∑i=1

∂f

∂xi(Wt1, . . . ,Wtn)RH(ti , s).

Starting withX = R, we are define the derivative of a real valued smooth func-tional F , then, takingX = H⊗n−1

H , we can define, inductively, thenth derivativeof F by ∇(n)F = ∇(∇(n−1)F ). The directional derivative ofF ∈ S(X) in thedirectionRHη ∈ HH is given by:

(∇F,RHη)HH= d

dtF (w + t.RHη))

∣∣∣∣t=0

(14)

and from the Cameron–Martin theorem, we see that∇F depends only on theequivalence classes with respect toPH and

EH [(∇F,RHη)HH] = EH [F 〈w, η〉]

which implies the closability of∇ and its iterates. We can also define Sobolevspaces using the operator∇ and its iterates as in the finite dimensional case, thisdefinition is equivalent to Definition [4.4]; this is due to the following inequalities

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 189

of P.A. Meyer: forp > 1 andk ∈ Z, there existap,k,H andAp.k,H such that, foreveryF ∈ S,

ap,k,H‖|∇(k)F |HS‖Lp(PH) 6 ‖F‖p,k,H6 Ap,k,H (‖F‖Lp(PH) + ‖|∇(k)F |HS‖Lp(PH)), (15)

where|∇(k)F |HS stands for the Hilbert–Schmidt norm of∇(k)F : if {ηn, n ∈ N} isan orthonormal basis ofH⊗kH ⊗X,

|∇(k)F |2HS =∞∑n=0

(∇(k)F, ηn)2H⊗kH ⊗X.

As a consequence,∇ can be extended as a continuous linear operator fromDHp,k(X)toDHp,k−1(HH ⊗X) for anyp > 1 andk ∈ Z.

DEFINITION 4.6 (Divergence or Skohorod integral). LetδH be the formal adjointof ∇ with respect toPH : ∀F ∈ S, ∀u ∈ S(HH),

EH [FδHu] = EH[(∇F, u)HH

]. (16)

Since∇ has continuous extensions,δH has a continuous linear extension fromDHp,k(HH) toDHp,k−1 for anyp > 1 and anyk ∈ N.

REMARK 4.2. ForF ∈ S(W ∗) of the form (11) withk = 1, the divergence ofFis defined by

(δHF)(w) = f (〈l1, w〉, . . . , 〈ln, w〉)〈x,w〉W∗,W

−n∑i=1

∂f

∂xi(〈l1, w〉, . . . , 〈ln, w〉) < RH(li), RH (x) >HH

.

Takec = εt , f = 1, we get

Wt = δH (RH(εt )) = δH (KH(KH(t, .))).Moreover, forη ∈ W ∗, we haveδH (RHη) = 〈η,w〉W∗,W almost surely. Further-more, we have

EH [FδHu] = EH [(∇F, u)HH]

for anyDHp,k+1, u ∈ DHq,−k(HH) for anyp > 1, p−1 + q−1 = 1 andk ∈ Z.We recall several identities valid in any Wiener space. LetF,F1, . . . , Fn be in

D∞,G1,G2 in D∞(HH), f in the Schwartz space ofRn andT ∈ D−∞(HH),

∇f (F1, . . . , Fn) =n∑i=1

∂if (F1, . . . , Fn)∇Fi,

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190 L. DECREUSEFOND AND A.S. ÜSTÜNEL

δH (∇F) = −LHF,

δH (FT ) = FδHT −∇T F, (17)

EH [δH (G1)δH (G2)] = EH [(G1,G2)HH] + EH [trace(∇G2 ◦ ∇G1)], (18)

∇G2(δHG1) = (G1,G2)HH+ δH (∇G2G1)+ trace(∇G2 ◦ ∇G1). (19)

PROPOSITION 4.1.For u ∈ HH , let3u1

def= exp(δHu− 1/2‖u‖2HH), we have

JHn 3u1 =

1

n!δ(n)H u

⊗n.

More generally forF ∈ ∪k∈ZDH2,k,

JHn F =1

n!δ(n)H (EH [∇(n)F ]).

Proof.By the definition (14) of∇, we have for any polynomialG onW,λ,

EH [G(w + λu)] =+∞∑n=0

λn

n!EH [(∇(n)G, u⊗n)H⊗nH ]

=+∞∑n=0

λn

n!EH [G.δ(n)H u

⊗n]

On the other hand, by the Cameron–Martin Theorem 4.1,

EH [G(w + λu)] =+∞∑k=0

EH [GJHk (exp(λδHu− λ2

2‖u‖2HH

))].

Using the generating functions of Hermite polynomials (see e.g. [11]), we see that

exp

(λδHu− λ

2

2‖u‖2HH

)=+∞∑n=0

Hn

(δHu√

2‖u‖HH

) ‖u‖nHH√2nn! λ

n,

whereHn is thenth Hermite polynomial. SinceHn(

δH u√2‖u‖HH

)belongs toCn,

EH [G(w + λu)] =+∞∑k=0

EH [G.JHk (F )]λk.

The result follows by identification of the two power series.

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 191

This result can also be written

JHn 3u1 =

1

n!δ(n)H (EH [∇(n)3u

1]).

Since the linear combinations of Wick exponentials (i.e., exponentials of the formexp(δHh − 1

2‖h‖2HH)) are dense inDH∞ andJHn is a continuous operator, the result

follows by density for anyF ∈ ∩k∈NDH2,k. Now by duality the formula also holdsfor F in the dual of the latter space, i.e., forF ∈ ∪k∈ZDH2,k. 2

4.1. RELATIONS BETWEENδh AND OTHER STOCHASTIC INTEGRALS

As for any Gaussian process, the Wiener integral with respect to the fractionalBrownian motion is usually defined as the linear extension fromHH in L2(PH) ofthe isometric map:

dw:RH(ti, .) 7→ Wti .

SinceKH(L2([0,1])) = HH , one could also define a Wiener integral for determ-inistic integrand belonging toL2([0,1]) by considering the linear extension fromL2([0,1]) toL2(PH) of the isometry:

∂w:KH(ti, .) 7→ Wti . (20)

WhenH = 1/2,K1/2(t, .) = 1[0,t ] so that we have:

∂w(h) = limn→+∞

2n−1∑i=0

hi2−n(W(i+1)2−n −Wi2−n), (21)

for any continuoush. From (20), it is clear that (21) does not hold any more whenH 6= 1/2. Thus there exist at least two different approaches to define a stochasticintegral with respect to the fractional Brownian motion: one approach consists ofusing the Skohorod integral which is defined for any Gaussian process, the secondapproach uses Riemann sums similar to the right-hand side of (21) see [3, 6, 9]. Theresulting integrals will have more or less similar properties to stochastic integralsdefined with respect to semi-martingales, however none of them will be completelyadequate to construct a full stochastic calculus.

1. We define stochastic integral of first type as:∫ 1

0usδHWs

def= δH (KH u)

for any u such thatKH u ∈ DH−∞(HH).

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192 L. DECREUSEFOND AND A.S. ÜSTÜNEL

2. ForH > 1/2, if we do not identifyHH and its dual butL2([0,1]) and itsdual, we have the following diagram:

W ∗i∗H - H∗H

K∗H- L2([0,1]) KH - HHiH - W

⋃‖

⋂W ∗

i∗1/2- H∗1/2K∗1/2- L2([0,1]) K1/2- H1/2

i1/2 - W.

Thus it is meaningful to define a stochastic integral of second type by:∫ 1

0us◦dWs =

∫ 1

0(K∗H u)(s)δHWs = δH (KHK∗H u),

whereKH = K−11/2KH , for u such thatKHK∗H u ∈ DH−∞(HH).

PROPOSITION 4.2.WhenH > 1/2, the stochastic integral of second type coin-cides with the stochastic integral defined by Riemann sums. We have the followingidentity provided thatu is deterministic and both sides exist:∫ 1

0us◦dWs = lim

|πn|→0

∑tni ∈πn

utni (Wtni+1−Wtni

). (22)

Whenu ∈ DH2,1(L2([0,1])) and trace(∇KHK∗H u) is well defined, we have:∫ 1

0us◦dWs = lim

|πn|→0

∑tni ∈πn

utni (Wtni+1−Wtni

)− trace(∇KHK∗H u). (23)

Proof. Note thatK∗1/2εt = 1[0,t ] andK∗Hεt = KH(t, .), thus K∗H (1[0,t ]) =KH(t, .). If u is of the form

u(s) =n∑i=1

ui1(ti ,ti+1](s),

with ui ∈ DH2,1, we have by (17),

δH (KHK∗H u) =n∑i=1

δH (uiKH (KH(ti+1, .)−KH(ti, .)))

=n∑i=1

uiδH (KH (KH(ti+1, .)−KH(ti, .)))

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 193

−∫ 1

0∇s ui(KH (ti+1, s)−KH(ti, s))ds

=n∑i=1

ui(Wti+1 −Wti )− trace(∇KH ◦K∗H u).

Since∇u = 0 whenu is deterministic, the two results follow by a limiting proced-ure when both sides of the last relation converge. 2In view of (23), it seems sensible to define a Stratonovitch integral by:

∫ 1

0us dWs =

∫ 1

0us◦dWs + trace(∇KHK∗H u),

for anyu such that the right-hand side of the last equation is meaningful. Note thatas it is shown in (23),

∫ 1

0us dWs = lim|πn|→0

∑tni ∈πn

utni (Wtni+1−Wtni

).

We will see below in the Itô formula that the Stratonovitch integral forH 6= 1/2does not behave as nicely as it does whenH = 1/2.

One can still have an integration by parts with◦dW if we use a damped Sobolev

derivative,

PROPOSITION 4.3.WhenH > 1/2, set

Dψdef= KH(KH(∇.ψ)).

We have:

EH

[∫ 1

0us◦dWs.ψ

]= EH [(KH u,Dψ)HH

], (24)

and

Df (Wt1, . . . ,Wtn)(s) =n∑i=1

∂f

∂xi(Wt1, . . . ,Wtn)

∫ s

0KH(s, r)

∂RH (ti , r)

∂rdr.

Note thattrace(∇KHK∗H u) = trace(DKHu), when both of the traces exist.

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194 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Proof.For any convenientu andψ ,

EH

[∫ 1

0us◦dWs.ψ

]= EH

[∫ 1

0(K∗H u)(s)δHWs.ψ

]

= EH

[∫ 1

0(K∗H u)(s)∇sψ ds

]

= EH

[∫ 1

0u(s)KH (∇.ψ)(s)ds

]= EH [(KH u,Dψ)HH

]The proof of the second part of the proposition is simply the application of thedefinitions ofD and∇. 2DEFINITION 4.7. For anyH ∈ (0,1), we define the family{πHt , t ∈ [0,1]} oforthogonal projections inHH by

πHt (KHu)def= KH(u1[0,t ]), u ∈ L2([0,1]).

The second quantization0(πHt ) of πHt is an operator fromL2(PH) into itselfdefined by,

F =∑n>0

δ(n)F fn 7→ 0(πHt )(F )

def=∑n>0

δ(n)H ((π

Ht )⊗nfn).

From [4.1], we have, foru ∈ HH ,

0(πHt )(λu1) = exp(δh(π

Ht u)− 1

2‖πHt u‖2HH)

def= 3ut .

The bijectivity of the operatorKH has the following consequence:

THEOREM 4.3. F Ht = σ {δH (πHt u), u ∈ HH } ∨NH .

Proof.The definition ofF Ht says that it is equal to the completion of theσ -field

generated by random variables of the formδH (KH (KH(s, .))) with s 6 t . Thisamounts to say that

F Ht = σ {δH(KHu), u ∈ span{KH(s, .), s 6 t}} ∨NH .

Observe now that we can replace span{KH(s, .), s 6 t} by its closure inL2([0, t])without changingF H

t . Actually, if u is the limit inL2([0,1]) of a sequence(un)nof elements of span{KH(s, .), s 6 t} thenKHun converges toKHu in HH and thusδH (KHun) tends toδH (KHu) in L2(PH). HenceδH (KHu) belongs toF H

t and the

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 195

two σ -fields are thus equal. Now it turns out that span{KH(s, .), s 6 t} is total inL2([0, t]): if g belongs to the orthogonal complement of this space inL2([0, t]):

0= (g,KH (s, .))L2([0,t ]) = KHg(s) for all s 6 t,

so thatg ≡ 0 in L2([0, t]). The proof is finished by observing that the imageL2([0, t]) byKH is nothing but the spaceπHt (HH). 2THEOREM 4.4. For anyF in L2(PH),

0(πHt )F = EH [F |F Ht ],

in particular,

EH [Wt |F Hr ] =

∫ t

0KH(t, s)1[0,r](s)δHWs, and

EH [exp(δHu− 1/2‖u‖2HH)|F H

t ] = exp(δHπHt u− 1/2‖πHt u‖2HH

),

for anyu ∈ HH .Proof.Let {hn, n > 1} be a denumerable family of elements ofHH and letVn =

σ {δHhk,16 k 6 n}. Denote byπn the orthogonal projection on span{h1, . . . , hn}.For anyf bounded, for anyu ∈ HH , by the Cameron–Martin theorem we have

EH [3u1f (δHh1, . . . , δHhn)]

= EH [f (δHh1(w + u), . . . , δHhn(w + u))]= EH [f (δHh1+ (h1, u)HH

, . . . , δHhn + (hn, u)HH)]

= EH [f (δHh1(w + πnu), . . . , δHhn(w + πnu))]= EH [3πnu

1 f (δHh1, . . . , δHhn)],hence

EH [3u1|Vn] = 3πnu

1 . (25)

Choosehn of the formπHt (en) where{en, n > 1} is an orthonormal basis ofHH ,i.e., {hn, n > 1} is an orthonormal basis ofπHt (HH). By the previous theorem,∨n Vn = F H

t and it is clear thatπn tends pointwise toπHt , hence from (25) andmartingale convergence theorem, we can conclude that

EH [3u1|F H

t ] = 3πHut = 3u

t .

Moreover, foru ∈ HH ,

0(πHt )(3u1) = 3π

Hut

1 ,

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196 L. DECREUSEFOND AND A.S. ÜSTÜNEL

hence by density of linear combinations of Wick exponentials, for anyF ∈ L2(PH),

0(πHt )F = EH [F |F Ht ],

and the proof is completed. 2THEOREM 4.5 (cf [16]). Let F be inDH2,1, F is F H

t -measurable if and only if∇F = πHt ∇F .

Proof. Let F be anF Ht -measurable element ofDH2,1, denote by{utn, n > 0} an

orthonormal basis ofL2([0, t]) andV tn be theσ field generated by{δHKHuti, i 6

n}. Since∨nV tn = F H

t , the sequenceFn = EH [F |V tn ] converges toF in DH2,1 and

Fn = fn(δHKHut1, . . . , δHKHutn) wherefn is a real valued smooth function. SinceπHt KHu

tn = KHutn, it follows that forFn we have

∇Fn =n∑i=1

∂fn

∂xt(δHKHu

t1, . . . , δHKHu

tn)KHu

tn

=n∑i=1

∂fn

∂xt(δHKHu

t1, . . . , δHKHu

tn)π

Ht KHu

tn = πHt ∇Fn,

hence∇F = πHt ∇F by a limiting procedure. In the converse direction, remarkthat it is sufficient to prove thatTsF is F H

t -measurable for anys > 0, where(Ts)sis the Ornstein–Uhlenbeck semigroup (see Definition [4.2]). It is easy to see that

∇TsF = ε−sTs∇F = ε−sTsπt∇F = πt e−sTs∇F = πt∇TsF.Iterating this relation, we obtain

∇(n)TsF = (πHt )⊗n∇(n)TsF.Moreover, from the Wiener chaos expansion, we have, for anyu ∈ HH ,

E[TsF (w + u)] =∞∑n=0

1

n!(E[∇(n)TsF ], u⊗n)H⊗nH

=∞∑n=0

1

n!(πH⊗nt E[∇(n)TsF ], u⊗n)H⊗nH

= E[TsF (w + πHt u)].On the other hand, from the Cameron–Martin formula,

EH [TsF exp(δHu− 12‖u‖2HH

)] = EH [TsF (w + u)]= EH [TsF (w + πtu)] = EH [TsF · EH [exp(δHu− 1

2‖u‖2HH)|F H

t ]]

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 197

and this completes the proof since the linear combinations of Wick exponentialsare dense in all the spacesDHp,k. 2PROPOSITION 4.4.A processu(w, t) is {F H

t , t ∈ [0,1]}-adapted iff itsdensityprocessu defined by:

u(w, t) =∫ t

0KH(t, s)u(w, s)ds

is {FHt , t ∈ [0,1]}-adapted.Proof.It is obvious that adaptedness ofu adapted entails thatu is adapted. In the

converse direction, note thatu(w, t) = (K−1H u(w, .))(t) and from Theorem [2.1]

we see that all the quadratures involved in the computation of(K−1h u)(t) have their

support in[0, t]. Hence, the adaptedness ofu entails the adaptedness ofu. 2THEOREM 4.6 (Itô–Clark representation formula).For anyF ∈ DH2,1,

F − EH [F ] =∫ 1

0EH [K−1

H (∇F)(s)|F Hs ]δHWs

= δH (KH (EH [K−1H (∇F)(·)|F ])).

Proof.From the chaos expansion (4.1), we know that for anyu ∈ HH ,

3u1 = 1+

∫ 1

0us3

us δHWs.

Moreover, linear combinations of Wick exponentials are dense inL2(PH) hence itis sufficient to prove that for anyu ∈ HH any centeredF ,

EH [F3u1] = EH

[∫ 1

0EH [K−1

H (∇F)(s)|F Hs ]δHWsδ(KH (u3

u))

].

Integrating by parts and since{3us , s ∈ [0,1]} is and adapted process, we have:

EH

[∫ 1

0EH [K−1

H (∇F)(s)|F Hs ]δHWs.δH (KH (u3

u))

]

= EH

[∫ 1

0EH [K−1

H (∇F)(s)|F Hs ]us3u

s ds

]

= EH

[∫ 1

0EH [K−1

H (∇F)(s)us3us |F H

s ds]]

= EH

[∫ 1

0K−1H (∇F)(s)us3u

s ds

]= EH [F.δH (KH (u3u))] = EH [F(3u

1 − 1)] = EH [F3u1],

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198 L. DECREUSEFOND AND A.S. ÜSTÜNEL

as it was required. 2THEOREM 4.7. For anyu adapted and inL2 (W ;HH ), the process{Mt = δH(πHt u), t ∈ [0,1]} is a square integrable martingale, i.e.,

EH [δHu|F Ht ] = δH (πHt u) =

∫ t

0(K−1

H u)(s)δHWs,

whose Doob–Meyer process ist 7→ ∫ t0 (K

−1H u) (s)2 ds. In particular, for v ∈ L2

(W ;L2([0,1])) adapted,

t 7→∫ t

0v(s)δHWs is a martingale.

Proof.Let u be adapted and belong toD∞(HH), for anyv ∈ HH , sinceπHt is aprojector,

EH [EH [δHu|F Ht ]3v

t ] = EH [δHu.3vt ]

= EH [(u,∇3vt )HH] = EH [(u, πHt v)HH

3vt ]

= EH [(πHt u, πHt v)HH3vt ] = EH [δH (πHt u)3v

t ].Moreover, for anyv ∈ HH such thatv = (IdHH

− πHt )v, we have

(∇δHπHt u, v)HH= (πHt u, (IdHH

− πHt )v)HH

+δH ((∇πHt u, (IdHH− πHt )v)HH

) = 0,

since by Theorem [4.5],∇πHt u = πHt ∇πHt u andπHt is an orthogonal projection.By density, we see that

EH [δHu|F Ht ] = δH (πHt u),

for anyu in DH−∞,a(HH).For anyt ∈ [0,1] and anyφ ∈ DH∞,

EH [(δH (πHt u)2− ‖πHt u‖2HH)φ]

= EH [(πHt u,∇(δH (πHt uφ)))HH] − EH [‖πHt u‖2HH

φ]= EH [(πHt u,∇φ)HH

] + EH [δH (∇πHt uπHt u)φ]= EH [(∇(2)φ, πHt u⊗ πHt u)HH⊗HH

] + EH [(∇(πHt u,∇φ)HH, πHt u)HH

].

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 199

Let 06 s 6 t 6 1 andφ be anF Hs -measurable element ofDH∞. Theorem [4.5]

implies that(∇φ, v)HH= (∇φ, πHs v)HH

for anyv. Hence,

EH [(δH (πHt u)2− ‖πHt u‖2HH)φ]

= EH [(∇(2)φ, πHs u⊗ πHs u)HH⊗HH] + EH [(∇(πHs u,∇φ)HH

, πHs u)HH]

= EH [(δH (πHs u)2 = ‖πHs u‖2HH)φ],

thus{δH (πHt u)2− ‖πHt u‖2HH, t ∈ [0,1]} is a martingale. 2

As a corollary we have aconstructiveproof of the Levy–Hida representation the-orem:

THEOREM 4.8 (Levy–Hida representation).

1. The process

B = {Bt def= δH (πHt KH1), t ∈ [0,1]}

is aPH -standard Brownian motion whose filtration is equal to{F Ht , t ∈ [0,1]}.

2. If we denote bydB the Itô integral with respect to this Brownian motion, wehave for any adaptedu,PH -almost surely,∫ t

0usδHWs =

∫ t

0us dBs, for all t.

Proof. B is a Gaussian process whose covariance kernel is given (see Corol-lary [4.7]) by

EH [BsBt ] = (πsKH1, πtKH1)HH=∫ 1

01[0,t ](u)1[0,s](u)du = min(s, t).

The equality of the filtrations follows simply from Proposition [4.3].Let u be of the form

u(w, s) =n∑i=1

ui1(ti ,ti+1](s),

where for anyi, ui is square-integrable andF Hti

-measurable. If for anyi, ui belongstoDH2,1 we have by (17)

∫ 1

0usδHWs =

n∑i=1

ui(Bti+1 − Bti )−∫ 1

0∇rui1(ti ,ti+1](r)dr.

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200 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Sinceui is F Hti

-measurable, the last integrand is zero by Theorem [4.5]. By alimiting procedure.∫ 1

0usδHWs =

n∑i=1

ui(Bti+1 − Bti ),

even whenui is only square integrable and isF Hti

-measurable. Moreover, by The-orem [4.7],

EH

[(∫ 1

0usδHWs

)2]= EH

[∫ 1

0u2s ds

].

On the other hand, the Itô stochastic integral ofu with respect toB is by definitiongiven by:

n∑i=1

ui(Bti+1 − Bti ).

By continuity of δH and dB, it follows that the stochastic integrations with re-spect toδHW or to dB coincide on the set of adapted processes which belong toL2(W ;HH ). 2Since∫ 1

0EH [∇sF |F H

s ]δHWs =∫ 1

0EH [∇sF |FHs ]dBs,

using Burkholder–Davis–Gundy inequality, we can show that:

COROLLARY 4.1 (cf [15], p. 42–44). Let F ∈ Dp,k, p > 1, k ∈ Z, then{EH [∇tF |F H

t ], t ∈ [0,1]} belongs toDp,k(L2([0,1])) and

F − EH [F ] =∫ 1

0EH [∇sF |F H

s ]δHWS

=∫ 1

0EH [∇sF |F H

s ]dBs.

where fork < 0, EH [F ] is to be interpreted as〈F,1〉 and∫ 1

0EH [∇sF |F H

s ]δHWs is defined asδH (KHEH [∇F |F H ]).

The following is the updated martingale representation theorem of K. Itô:

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 201

COROLLARY 4.2. Every(PH , {F Ht , t ∈ [0,1]}) square integrable martingaleM

can be written as

M0+ δH (πHt u),where

ut = EH [∇tM1|F Ht ].

THEOREM 4.9 (Girsanov theorem).Let u = KH u be an adapted process inL2(W ;HH ) such that

EH [3u1] = 1, (26)

let Pu be the probability defined by

dPudPH

∣∣∣∣F Ht

= 3ut = exp(δHπ

Ht u− 1

2‖πHt u‖2HH).

The law of the process

{Wt −∫ t

0KH(t, s)us ds, t ∈ [0,1]} underPu

is the same as the law of the canonical processW underPH . In other words, foranyv = KH v in HH , the law of the process{∫ t

0KH(t, s)vsδFWs −

∫ t

0KH(t, s)us vs , t ∈ [0,1]

}underPu

is the same as the law of the process{∫ t

0KH(t, s)vsδHWs, t ∈ [0,1]

}underPH .

Proof.Since

Wt −KH u(t) =∫ t

0KH(t, s)(dBs − us ds),

from the classical Girsanov theorem, we have

EH

[F(w −KHu)exp

(∫ 1

0us dBs − 1

2‖u‖2HH

)]= EH [F ],

for anyF ∈ Cb(W). 2

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202 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Since we have reduced the Girsanov problem of fBm to that of the ordinary Brownianmotion, we can make the full use of the usual Novikov condition to ensure theuniform integrability of3u. Namely, it is sufficient that

EH [exp12‖u‖2HH

] < +∞

for (26) to hold.

Application: Consider the processX defined by

Xt = θt +WHt

and let PX denote its law. We aim to estimateθ through the observation of asample-path ofX over [0, t]. Let φ be such that(KHφ)(t) = t , by the Cameron–Martin Theorem and since for deterministicu, (δHu)(w + KHφ) = (δHu)(w) +(u,KHφ)HH

dPXdPH

∣∣∣∣F Ht

(w) = EH

[exp(θδH (KHφ)(w)− θ

2

2‖φ‖2

L2)|F Ht

]

= exp(θδH (πHt KHφ)(w)−

θ2

2‖πHt φ‖2L2),or

dPHdPX

∣∣∣∣F Ht

(w) = exp(−θδH (πHt KHφ)(w)+θ2

2‖πHt φ‖2L2)|, i.e.,

EH [F(w)] = EH

[F(x(w)) exp(−θδH (πHt KHφ)(X(w))+

θ2

2‖πHt φ‖2L2)

].

Hence thePH maximum likelihood ratio estimateθt of θ is

θt = 1

‖πHt φ‖2HH

δH {πHt KHφ)(X(w)).

From Theorem [2.1], we have

φ(s)def= 0(3/2−H)

0(2− 2H)s1/2−H .

Hence,

θt = 22−2H0(2−H)√πt2−2H

δH(πHt KH s

1/2−H)(X(w)).

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 203

5. The Itô Formula

Hereafter we assume thatH is greater than 1/2. In this case, the fractional Brownianmotion has a zero quadratic variation hence it is a Dirichlet process. It is wellknown that for such processes there exists an Itô formula (see [7]) of the form:

F(Wt) = F(W0)+∫ t

0F ′(Ws)dWs, (27)

where the stochastic integral dWs is defined as the limit of Riemann sums as informula (22) andF is of classC2. In fact, we improve, in our situation, the resultsknown for Dirichlet processes in the sense that a somewhat explicit expressionof the right-hand side of (27) is given. The processes of the form (28) have beenchosen because they constitute a class which is stable with respect to absolutelycontinuous changes of probability measures – see Theorem [4.9].

Let ρ(s) = s1−2H and denote byHH,ρ the image ofL2ρ([0,1]) = {h: [0,1] →

R,∫ 1

0 h2(s)ρ(s)ds < +∞} underKH . Note that whenH > 1/2, L2

ρ([0,1]) ⊂L2([0,1]) becauseρ(s) > 1 for anys ∈ [0,1]. The spaceHH,p is endowed withthe norm induced byL2

ρ([0,1]), i.e.,

‖u‖HH,ρ= ‖K−1

H u‖L2ρ.

The weighted Sobolev spaceDH2,k,ρ is the set of elementsF of DH2,k such that foranyn 6 k, thenth Gross–Sobolev derivative ofF belongs toH⊗H,ρ and the normof F in this space is defined by

‖F‖22,k,H,ρ = ‖F‖2L2(PH )+

k∑i=1

EH [‖∇(i)F‖2H⊗iH,ρ ].

Before the proof, we give two lemmas for later use.

LEMMA 5.1. Letu(w, s) be such thatEH [∫ 1

0 |u(w, s)|ds] < +∞ thent → EH[I α0+u(t)] is continuous.

Proof.SinceI α0+ is a positive linear operator,

|EH [I α0+u(t + ε)− I α0+u(t)]| 6 (I α0+EH [|u|])(t + ε)− (I α0+EH [|u|])(t).Hence by the remark following Proposition [2.1], the right-hand side of the lastinequality converges to 0 asε tends to 0. 2LEMMA 5.2. Let(Yε)ε be a process which converges to0 in L2(PH) whenε tendsto 0. Leth(w, u) be inL2(L2([0,1]). Then

limε→0

EH

[ε−1Yε

∫ t+ε

t

h(u)du

]= 0 dt a.s.

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204 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Proof.Using the Cauchy–Schwarz inequality,

EH

[ε−1Yε

∫ t+ε

t

h(u)du

]2

6 2EH [Y 2ε ]{EH

[(∫ t+ε

t

h(u)− h(t)duε

)2]+ 2EH [h(t)2]

}

6 2EH [Y 2ε ]{ε−1

∫ t+ε

t

EH [(h(u)− h(t))2]du+ 2EH [h(t)2]}.

The second term of the previous sum is finite for almost everyt and the firstconverges to 0 asε tends to 0, hence all of the right-hand side converges to 0for almost everyt . 2THEOREM 5.1. SupposeH > 1/2, let F :R 7→ R be twice differentiable withbounded derivatives andX be a process of the form

Xt(w) = X0(w)+ (KHξ(w))(t)+ δH (KH(σ (w, .)KH(t, .)))(w)

= X0(w)+∫ t

0KH(t, s)ξ(w, s)ds +

∫ t

0KH(t, s)σ (w, s)δHWs, (28)

whereX0 belongs toDH2,1,ρ, ξ is inDH2,1,ρ(L2ρ([0,1])) andσ is inDH2,2,ρ(L2

ρ([0,1])).We have

F(Xt)− F(X0) =∫ t

0IH−1/2t− (F ′(Xu)uH−1/2)(s)s1/2−Hξ(s)ds

+∫ t

0s1/2−Hσ (s)IH−1/2

t− (uH−1/2F ′(Xu))(s)δHWs

+∫ t

0IH−1/2t− (F ′′(Xu)uH−1/2)(s)s1/2−Hσ (s)∇sX0 ds

+∫ t

0IH−1/2t− (F ′′(Xu)uH−1/2KH(∇sξ )(u))(s)

× s1/2−Hσ (s)ds

+∫ t

0IH−1/2t− (F ′′(Xu)uH−1/2KH(u, s))(s)s

1/2−Hσ (s)2 ds

+∫ t

0IH−1/2t− (F ′′(Xu)uH−1/2

∫ 1

0∇sσ (r)KH(u, r)δHWr)(s)

× s1/2−Hσ (s)ds. (29)

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 205

Proof. Let us give the main idea of the proof: as a first step we shall supposethatX0, ξ, σ are smooth processes in the sense that they have bounded Sobolev–Gross derivatives which are continuous with respect to all their parameters, i.e.,∇uX0, ∇uσ (s), ∇uξ(s) and∇u,vσ (s) are bounded and continuous with respect tow, u, v ands. Then we use the fundamental theorem of differential calculus whichsays that

EH [F(Xt)ψ] = EH [F(X0)ψ] +∫ t

0

d

dsEH [F(Xs)ψ]ds,

whereψ is in D∞. Afterwards, we rewrite the above identity using the stochasticintegration by parts formula (16) and Fubini’s theorem to obtain the claimed ex-pression.

(1) In a first step, we assume that we have the following additional hypothesis:F

is C2 with bounded derivatives,∇uX0, ∇uσ (s), ∇uξ(s) and∇u,vσ (s) are boundedand continuous with respect tow, u, v ands. Letψ be inDH∞ with bounded Gross–Sobolev derivatives,

EH [(F (Xt+ε)− F(Xt))ψ] = EH [F ′(Xt )(Xt+ε −Xt)ψ]

+EH [(Xt+ε −Xt)2∫ 1

0F ′′(uXt + (1− u)Xt+ε)(1− u)du.ψ] = A1+ A2.

Let us first considerA1:

A1 = EH [F ′(Xt )ψ(KHξ(t + ε)−KHξ(t))]+EH [F ′(Xt)ψδH (KH (σ {KH(t + ε, .)−KH(t, .)}))] = B1+ B2.

Sinceξ is bounded,t 7→ (KHξ)(t) is differentiable for eachw and

(KHξ)′(t) = tH−1/2I

H−1/20+ (u1/2−Hξ(u))(t).

Hence, by the dominated convergence theorem,

limε→0

ε−1B1 = EH [F ′(Xt)tH−1/2IH−1/20+ (u1/2−Hξ(u))(t).ψ]. (30)

Because of the regularity ofX0, ξ andσ , it is clear thatXt has a Gross–Sobolevderivative for everyt and moreover (cf. (19)):

∇uXt = ∇uX0+∫ t

0KH(t, s)∇uξ(s)ds

+σ (u)KH(t, u)+∫ t

0∇uσ (s)KH (t, s)δHWs.

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206 L. DECREUSEFOND AND A.S. ÜSTÜNEL

Hence, we can write

B2 = EH

[∫ 1

0σ (u)(KH(t + ε, u)−KH(t, u))∇(F ′(Xt)ψ)du

]= EH [KH(σ (·)∇(F ′(Xt)ψ))(t + ε)−KH(σ (·)∇(F ′(Xt )ψ))(t)].

The additional hypothesis imply thatt 7→ KH(σ (·)∇(F ′(Xt)ψ))(t) is continu-ously differentiable and by the dominated convergence theorem, we have:

limε→0

ε−1B2 = EH [tH−1/2IH−1/20+ (u1/2−Hσ (u)∇u(F ′(Xt )ψ))(t)].

Consider now the second-order term. SinceF ′′ is bounded

A2 ' cEH [(Xt+ε −Xt)2.ψ].Furthermore, by the reasoning used to obtain (30),

EH [(KHξ(t + ε)−KHξ(t))2] = o(ε2)

so that, it is sufficient to look at the term involving the stochastic integral:

EH [F ′′(Xt)ψδH (KH(σ {KH(t + ε, .)−KH(t, .)}))2]

= EH

[F ′′(Xt)ψ

(∫ t

0σ (u)(KH(t + ε, u)−KH(t, u)

)δHWu

−∫ t+ε

t

σ (u)KH(t + ε, u)δHWu)2

].

= EH

[F ′′(Xt)ψ

(∫ t

0σ (u)(KH(t + ε, u)−KH(t, u))δHWu

)2]

+EH

[F ′′(Xt)ψ

(∫ t+ε

t

σ (u)KH (t + ε, u)δHWu

)2]

+2EH

[F ′′(Xt )ψ

(∫ t

0σ (u)(KH(t + ε, u)−KH(t, u))δHWu

×∫ t+ε

t

σ (u)KH(t + ε, u)δHWu

)]= C1+ C2+ 2C3.

By (18) and the Cauchy–Schwarz inequality, we can upperboundC1 by

|C1| 6 ‖F ′′‖∞‖ψ‖∞EH

[∫ t+ε

0σ (u)2(KH(t + ε, u)−KH(t, u))2 du

]

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 207

+‖F ′′‖∞‖ψ‖∞EH

[∫ t

0

∫ t

0∇uσ (s)∇sσ (u)(KH(t + ε, u)−KH(t, u))

(KH (t + ε, s)−KH(t, s))duds]

6 C

∫ t

0(KH(t + ε, u)−KH(t, u))2 du = CVHε2H.

By the same way, there exists a constantc such that

|C3| 6 c

∫ t+ε

t

(KH(t + ε, u)−KH(t, u))KH (t + ε, u)du

+c(∫ t

0(KH (t + ε, u)−KH(t, u))du

)(∫ t+ε

t

KH (t + ε, u)du

).

As toC2, we have

|C2| 6 cEH

[∫ t+ε

t

σ (u)2KH(t + ε, u)2 du

]

+cEH[∫ t+ε

t

∫ t+ε

t

∇uσ (u)∇uσ (s)KH(t + ε, u)KH (t + ε, s)duds

],

with another constantc. When divided byε, the right-hand-side of the last equationgoes to 0 becauseKH(t, t) = 0. We have proved so far that, under the additionalsmoothness hypothesis, we have

d

dtEH [F(Xt)ψ] = EH [F ′(Xt)tH−1/2I

H−1/20+ (u1/2−Hξ(u))(t).ψ]

+EH [tH−1/2F ′(Xt )IH−1/20+ (u1/2−Hσ (u)∇uψ)(t)]

+EH [tH−1/2F ′′(Xt )IH−1/20+ (u1/2−Hσ (u)∇uX0)(t).ψ]

+EH

[tH−1/2F ′′(Xt)I

H−1/20+(

u1/2−Hσ (u)∫ t

0KH(t, r)∇uξ(r)dr

)(t).ψ

]+EH [tH−1/2F ′′(Xt )I

H−1/20+ (u1/2−Hσ (u)2KH(t, u))(t).ψ]

EH

[tH−1/2F ′′(Xt)I

H−1/20+(

u1/2−Hσ (u)∫ 1

0∇uσ (s)KH (t, s)δHWs

)(t).ψ

]. (31)

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208 L. DECREUSEFOND AND A.S. ÜSTÜNEL

(2) The second step is to prove that each of these terms is continuous withrespect tot in order to be able to integrate them over a finite interval. The firstthree terms are easily handled using Lemma [5.1]. We only give here the completeproof for the last term of the previous sum because it is the most difficult one,fourth and fifth terms are handled similarly: sincetH−1/2F ′′(Xt )ψ is bounded, it issufficient to prove that

t → EH

[IH−1/20+

(u1/2−Hσ (u)

∫ 1

0∇uσ (s)KH (t, s)δHWS

)(t)

]is continuous. Let us defineYt,u =

∫ 10 ∇u σ (s)KH (t, s)δHWs. We have:

|EH [IH−1/20+ (u1/2−Hσ (u)Yt+ε,u)(t + ε)− IH−1/2

0+ (u1/2−Hσ (u)Yt,u)(t)]|6 |EH [IH−1/2

0+ (u1/2−Hσ (u)(Yt+ε,u − Yt,u))(t + ε)]|+ |EH [IH−1/2

0+ (u1/2−Hσ (u)Yt,u)(t + ε)− IH−1/20+ (u1/2−Hσ (u)Yt,u)(t)]|. (32)

By (17),

EH [(Yt+ε,u − Yt,u)2] 6 EH

[∫ 1

0(∇uσ (s))2(KH(t + ε, s) −KH(t, s))2 ds

]

+EH

[∫ ∫|∇v,uσ (s)∇s,uσ (v)|(KH(t + ε, s)−KH(t, s))

(KH (t + ε, v)−KH(t, v))ds dv

]

6 C∫ 1

0(KH (t + ε, s)−KH(t, s))2 ds

6 C(RH(t + ε, t + ε)+ RH(t, t)− 2RH(t + ε, t)) 6 Cε2H−1,

whereC is a constant independent ofu. Hence,

|EH [IH−1/20+ (u1/2−Hσ (u)(Yt+ε,u − Yt,u))(t + ε)]|

6 CIH−1/20+ (u1/2−H)(t + ε).εH−1/2.

As to the last term of (32), using [5.1], it can be made as small as desired since∫ 1

0u1/2−HEH [σ (u)Yt,u]du 6 C‖σ‖∞

∫ 1

0u1/2−HRH(t, u)1/2 du < +∞.

Now, before we can relax the additional hypothesis, we apply the Fubini Theorem(to exchange expectations and integrals) and the fractional integration by parts (5).

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 209

For instance, for the second term of the right-hand-side sum, we obtain

EH

[∫ t

0sH−1/2F ′(Xt )I

H−1/20+ (u1/2−Hσ (u)∇uψ)(s)ds

]

= EH

[∫ t

0IH−1/2(uH−1/2F ′(Xu))(s)s1/2−Hσ (s)∇sψ ds

]

= EH

∫ t

0s1/2−Hσ (s)IH−1/2

t− (uH−1/2F ′(Xu))(s)δHWs

].

The other terms are transformed similarly. If we denote byRHS the right-hand-side of (29), we know that

EH [(F (Xt)− F(X0)− RHS)ψ] = 0,

for anyψ in DH∞ with bounded derivatives. Since both ofF(Xt) − F(X0) andRHS belong toL2(PH), we deduce by the density ofDH∞ in L2(PH) thatF(Xt)−F(X0) = RHS.

(3) The third step is to prove that the additional hypothesis can be relaxed.Actually, our aim is to show that givenXn

0, σn, ξn andFn converging respectively

toX0, σ, ξ andF in the respective normed spaces, the sequenceFn(Xnt )−Fn(Xn

0)−RHS(Fn,X

n) converges inL1H toF(Xt)−F(X0)−RHS(F,X). For this it is suf-

ficient to show thatEH [RHS(F,X)] can be bounded by a polynomial in‖F ′‖∞,‖F ′′‖∞, ‖X0‖2,1,H,ρ, ‖ξ‖DH2,1,ρ (L2

ρ([0,1])) and‖σ‖DH2,2,ρ (L2ρ([0,1])). For instance, for the

last term (29), we have

EH

[∫ t

0IH−1/2t−

(F ′′(Xu)uH−1/2

∫ 1

0∇sσ (r)KH(u, r)δHWr

)(s)s1/2−Hσ (s)ds

]

6 EH

[∫ t

0(s1/2−Hσ (s))2 ds1/2

]

×EH

[(∫ t

0IH−1/2t− (uH−1/2F ′′(Xu)

∫ 1

0∇sσ (r)KH(u, r)δHWr)(s)

)2

ds

]1/2

6 C‖F ′′‖∞‖σ‖L2(L2ρ)

×EH

[∫ t

0

(∫ t

0

∫ u

0∇sσ (r)KH (u, r)δHWr(u− s)H−3/2 du

)2

ds

]1/2

.

By the Jensen inequality applied to the measure(u − s)H−3/21[s,t ](u)du (whosetotal mass is(H − 1/2)−1(t − s)H−1/2), we upperbound the last expectation by a

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210 L. DECREUSEFOND AND A.S. ÜSTÜNEL

constant times the following integral:∫ t

0(t − s)H−1/2

∫ t

s

EH

[(∫ u

0∇sσ (r)KH (u, r)δHWr

)2](u− s)H−3/2 duds

6 EH

[∫ t

0(t − s)H−1/2

∫ t

s

∫ 1

0(∇sσ (r)r1/2−H)2 dr duds

]

+EH

[∫ t

0(t − s)H−1/2

∫ t

s

∫ 1

0

∫ 1

0|∇v,s(r)|2ρ(r)ρ(v)dr dv duds

]

6 EH

[∫ 1

0

∫ 1

0(∇sσ (r)(sr)1/2−H )2 dr ds

]

+EH

[∫ 1

0

∫ 1

0

∫ 1

0|∇v,sσ (r)|2ρ(r)ρ(v)ρ(s))dr dv ds

]6 C‖σ‖22,2,ρ,

where we have successively used (18), the upperbound (9) ofKH , the fact thatρ(s) > 1 for anys ∈ [0,1] and the Cauchy–Schwarz inequality. 2REMARK 5.1. Note that whenH = 1/2, the contribution of the second order termA2 is 1/2.

∫ t0 σ

2(s)ds which is the term corresponding to the square bracket in the

classical Itô formula. Still whenH = 1/2, IH−1/20+ = Id and (29) differs from

the Itô formula in [14] by only 1/2∫ t

0 σ2(s)ds. Moreover when the processesσ

andξ andX0 are{F Ht , t ∈ [0,1]}-adapted, all the terms involving Gross–Sobolev

derivatives vanish whenH = 1/2 but not whenH > 1/2 because of the derivativeof the kernelKH(t, .).

COROLLARY 5.1. LetH > 1/2 andf :R→ R be a twice differentiable functionwith bounded derivatives. Defineu(t, x) = EH [f (x +Wt)], we have:

∂tu(t, x) = HVHt2H−1 ∂

2

∂x2u(t, x).

Proof.Applying (31) withψ = 1, we obtain:

∂tu(t, x) = EH [F ′′(Xt)]tH−1/2IH−1/20+ (u1/2−HKH(t, u))(t).

Now, using Theorem [2.1], we see that for anyf ∈ L2([0,1]),IH−1/20+ (u1/2−Hf )(s) = s1/2−HI−1

0+ (KHf )(s).

Hence,

IH−1/20+ (u1/2−HKH(t, u))(t) = t1/2−H ∂RH

∂s(t, s)

∣∣∣∣s=t= HVHtH−1/2

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 211

which proves the result. 2Before we state the Itô formula for processes defined by the stochastic integral ofsecond kind, we need the following result:

LEMMA 5.3. For anyf ∈ L2ρ([0,1]), we have∫ 1

0(KHf )(s)

2 ds 6 c∫ 1

0f (s)2s1/2−H ds,

∫ 1

0(K∗Hf )(s)

2 ds 6 c∫ 1

0f (s)2s1/2−H ds,

∫ 1

0(KHK∗Hf )(s)

2 ds 6 c∫ 1

0f (s)2s1/2−H ds,

whereKH = K−11/2KH .

Proof.From Theorem (2.1), we know that

K∗Hf = x1/2−HIH−1/21− (xH−1/2f ).

SinceIH−1/21− is a positive operator, using Cauchy–Schwarz inequality,∫ 1

0(K∗Hf )(s)

2 ds =∫ 1

0s1−2HI

H−1/21− (uH−1/2f )(s)2 ds

6 c

∫ 1

0s1−2HI

H−1/21− (f )(s)2 ds

6 c

∫ 1

0s1−2H(1− s)H−1/2

∫ 1

s

(u− s)H−3/2f (u)2 duds

6 c

∫ 1

0f (u)2

∫ u

0s1−2H(u− s)H−3/2 ds du

6 c

∫ 1

0f (s)2s1/2−H ds,

wherec denotes any constant. The other inequalities are shown similarly. 2THEOREM 5.2. Assume that for anyt ∈ [0,1],

Xt = x0+∫ t

0ξs ds +

∫ t

0σs◦dWs,

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212 L. DECREUSEFOND AND A.S. ÜSTÜNEL

whereξ belongs toDH2,1,ρ(L2([0,1])) andσ ∈ DH2,2,ρ(L2ρ([0,1])). For anyF twice

differentiable with bounded derivatives, we have for anyt,PH -almost surely:

F(Xt) = F(X0)+∫ t

0F ′(xs)ξs ds +

∫ t

0F ′(Xs)σs

◦dWs,

+∫ t

0F ′′(Xs)DsXsσs ds, (33)

with

DtXt =∫ t

0Dtξs ds + (KHK∗Hσ )(t)+

∫ t

0Dtσs

◦dWs,

andDtφ is defined asKH(∇φ)(t) (see(24)).

Sketch of the proof.Formally, the proof follows the lines of the preceding one.The third step consists also of upper-bounding the expectation of the right-hand side of (33) by a polynomial in‖F ′‖∞, ‖F ′′‖∞, ‖ξ‖DH2,1,ρ (L2

ρ([0,1])) and‖σ‖DH2,2,ρ (L2

ρ([0,1])). For instance,

EH

[∣∣∣∣∫ t

0F ′(Xs)σs

◦dWs

∣∣∣∣]2

6 EH [|δHKHK∗H(F′(X)σ )|2]

6∫ 1

0K∗H(F

′(X)σ )(s)2 ds

+trace(∇KHK∗H(F′(X)σ ) ◦ ∇KHK∗H(F

′(X)σ ))

6 ‖F ′‖2∞‖K∗Hσ‖2L2 + ‖∇KHK∗H(F′(X)σ )‖2HS.

Moreover,

∇rK∗H (F ′(X)σ )(u) = K∗H (F′′(X)σ ∇rX)(s)

+K∗H (F′(X)∇rσ (·))(s).

Hence by Lemma [5.3], we have

EH

[∣∣∣∣∫ t

0F ′(Xs)σs

◦dWs

∣∣∣∣]2

6 c‖F ′‖2∞‖σ‖2DH2,1,ρ (L2ρ([0,1]))

+c‖F ′′‖2∞‖σ‖2L2ρ(‖σ‖2DH2,2,ρ ([0,1])) + ‖ξ‖

2DH2,1,ρ ([0,1]))),

wherec is a constant. The other terms are treated similarly. 2

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STOCHASTIC ANALYSIS OF THE FRACTIONAL BROWNIAN MOTION 213

THEOREM 5.3. Assume that for anyt ∈ [0,1],

Xt = x0+∫ t

0ξs ds +

∫ t

0σsdWs,

whereξ belongs toDH2,1,ρ(L2([0,1])) andσ ∈ D2,2,ρ(L2ρ([0,1])).

1. The integral∫ 1

0 DtXt dt is finite, and2. For anyF twice differentiable with bounded derivatives, we have for anyt,PH -

almost surely:

F(Xt) = F(X0)+∫ t

0F ′(Xs)ξs ds +

∫ t

0F ′(Xs)σsdWs

−∫ t

0F ′′(Xs)DsXsσsdWs.

Proof.By the computations we made in the previous proof, we have∫ 1

0(DtXt )

2 dt is finite.

The last part follows from the previous Itô formula. 2

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