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E l e c t ro n ic

Jo

u r n a l

o f

P

r o b

a b i l i t y Vol. 8 (2003) Paper no. 14, pages 1–31.

Journal URL

http://www.math.washington.edu/˜ejpecp/

Itˆo Formula and Local Time for the Fractional Brownian Sheet Ciprian A. Tudor1 and Frederi G. Viens2

1_{Laboratoire de Probabilit´es, Universit´e de Paris 6}

4, place Jussieu, 75252 Paris, France Email: tudor@ccr.jussieu.fr

and

2_{Dept. Statistics and Dept. Mathematics, Purdue University}

150 N. University St., West Lafayette, IN 47907-2067, USA Email: viens@purdue.edu

Abstract. Using the techniques of the stochastic calculus of variations for Gaussian pro-cesses, we derive an Itˆo formula for the fractional Brownian sheet with Hurst parameters bigger than 1/2. As an application, we give a stochastic integral representation for the local time of the fractional Brownian sheet.

Key words and phrases: fractional Brownian sheet, Itˆo formula, local time, Tanaka formula, Malliavin calculus.

1

Introduction

Let (Bα

t)t∈[0,1] be the fractional Brownian (fBm) motion with Hurst parameter α∈ (0,1).

Whenα= 1/2,B1/2 _{is the standard Brownian motion. Whenα6= 1/2, this process is not a}

semimartingale and in this case the powerful theory and tools of the classical Itˆo stochastic calculus cannot be applied to it. However, its special properties such as self-similarity, stationarity of increments, and long range dependence make this process a good candidate as a model for different phenomena. Therefore, a development of the stochastic calculus with respect to the fBm was needed. We refer to [1], [6], [9] and [11] for some approaches of this stochastic calculus.

The aim of this paper is to develop a stochastic calculus for the two-parameter fBm introduced in [3], also known as the fractional Brownian sheet. As the one-parameter fBm, the fractional Brownian sheet is not a semimartingale. But the Malliavin calculus for Gaussian processes can be adapted to it, Skorohod stochastic integrals can be defined, and it will allow us to derive an Itˆo formula when the Hurst parameters are bigger than 1/2 which extends the Itˆo formula for the two parameters martingales (see [10], [16] and [13] for a complete exposition of this topic). As an application we study the representation of the local time of the fractional Brownian sheet in terms of Skorohod stochastic integrals.

Our paper is organized as follows: Section 2 contains the basic notions of the Malli-avin calculus for Gaussian processes and the definition of the fractional Brownian sheet. In Section 3 we give an analogue of the Doob-Meyer decomposition for the square of a martin-gale, and an Itˆo formula for the fractional Brownian sheet. The interpretation of a specific new term appearing in the Itˆo formula is given in Section 4. Finally, as an application of this formula, Section 5 contains the study of the local time of the fractional Brownian sheet using stochastic integrals and an occupation time formula.

2

Preliminaries

2.1 The Malliavin calculus for Gaussian processes

Let T = [0,1]. Consider (Bt)t∈T a centered Gaussian process, with covariance E(BtBs) = R(t, s).

We consider the canonical Hilbert space _H associated with the process B, defined as the closure of the linear space generated by the function _{1[0,t], t ∈ T} with respect to

the scalar product _h1_[0,t],1_[0,s]i =R(t, s). Then the mapping 1_[0,t]→ Bt gives an isometry

between _H and the first chaos generated by_{Bt, t∈ T} and B(φ) denotes the image of a

elementφ_{∈ H}.

We can develop a stochastic calculus of variations, or a Malliavin calculus, with respect to B. For a smooth _H-valued functional F = f(B(ϕ1),· · · , B(ϕn)) with n ≥ 1 , f _∈C_b∞(Rn) and ϕ1,_{· · ·} , ϕn_{∈ H} we put

DB(F) =

n X j=1

∂f ∂xj

and DBF will be closable from Lp(Ω) into Lp(Ω;_H). Therefore, we can extendDB to the closure of smooth functionals on the Sobolev space Dk,p _{defined by the norm}

kF_kp_k,p=E_|F_|p+

k X j=1

kDB,jF_kp_Lp_(Ω;H⊗j₎.

Similarly, for a given Hilbert space V we can define Sobolev spaces of V-valued random variables Dk,p(V) (see [17]).

Consider the adjoint δB _{of D}B_{. Then its domain is the class ofu∈ L}2_{(Ω;H) such}

that

E_|hDBF, u_iH| ≤CkFk2 and δB(u) is the element of L2(Ω) given by

E(δB(u)F) =E_hDBF, u_iH

for everyF smooth. It is well-known that D1,2(_H) is included in the domain ofδB. Note that EδB(u) = 0 and the variance of the divergence operatorδB is given by

EδB(u)2=E_ku_k2_H+E_hDBu,(DBu)∗_iH⊗H (1) where (DB_u)∗ _{is the adjoint ofD}B_{u in the Hilbert space H ⊗ H. This last identity implies}

that

EδB(u)2 _≤E_ku_kH2 +E_kDBu_k2_H⊗H. (2) We will need the property

F δB(u) =δB(F u) +_hDBF, u_iH (3)

ifF _∈D1,2 and u_∈Dom(δB) such thatF u_∈Dom(δB).

2.2 Representation of fBm

Let nowB =Bα_{be the fractional Brownian motion with parameterα∈(0,1). This process}

is centered Gaussian,B0= 0 and R(t, s) = 1

2

³

s2α+t2α_{− |}t₋s_|2α´. (4) We know thatB admits a representation as Wiener integral of the form

Bt= Z t

0

K(t, s)dWs, (5)

whereW =_{Wt, t∈T}is a Wiener process, and K(t, s) is the kernel

K(t, s) =dα(t−s)α−

1

2 +sα−12F₁

µ t s

¶

dα being a constant and

F1(z) =dα µ

1 2 −α

¶ Z z−1 0

θα−32

³

1₋(θ+ 1)α−12

´

dθ. (7)

This kernel satisfies the condition (see [15]):

∂K

∂t (t, s) =dα(α−

1 2)(

s t)

1

2−α(t−s)α−32. (8)

We will denote by _Hα its canonical Hilbert space and byDBα and δBα the corresponding Malliavin derivative and Skorohod integral with respect toBα.

Fix now α_∈(1₂,1) and define the operatorK⋆ _{from the set of step functions on T}

(denoted by _E) to L2(T) by

(K⋆f)(s) =

Z 1 s

f(r)∂K

α

∂r (r, s)dr.

It was proved in [1] that the space _Hα of the fBm can be represented as the closure of _E with respect to the norm _kf_kHα =kK⋆fkL2_(T) and K⋆ is an isometry betweenHα and a

closed subspace of L2(T). This fact implies the following relation between the Skorohod integration with respect toBα and W, the Wiener process:

δBα(u) =δW(K⋆u) (9)

for any_Hα_{-valued random variableu inDom(δ}Bα

). We will callδBα

(u) the Skorohod inte-gral of the processuwith respect toB. Sinceα >1/2, we have the following representation of the scalar product in _Hα:

hf, g_iHα =α(2α−1)

Z 1 0

Z 1 0

f(a)g(b)_|a₋b_|2α−2dadb. (10) This formula holds as long as the integral above is finite whenf and g are replaced by _|f_|

and _|g_|.

2.3 Fractional Brownian sheet

Consider (W_s,tα,β)_s,t∈[0,1] a fractional Brownian sheet with parameters α, β _∈ (0,1). This process is Gaussian, it starts from 0 and its covariance is

E³W_t,sα,βW_u,vα,β´=Rα(t, u)Rβ(s, v) = 1

2

¡

t2α+u2α_{− |}t₋u_|2α¢1

2

³

s2β+v2β _{− |}s₋v_|2β´.

This process self similar and with stationary increments and it admits a continuous version. We refer to [3] for the basic properties of Wα,β. The fractional Brownian sheet with Hurst parameters α, β_∈(0,1) can be defined as (see [4])

W_s,tα,β =

Z t 0

Z s 0

where (Wu,v)u,v∈[0,1] is the Brownian sheet.

The canonical Hilbert Space _Hα,β _{of the fractional Brownian sheet is the closure}

of linear space generated by the indicator functions on [0,1]2 with respect to the scalar product

One can develop a Malliavin stochastic calculus with respect to the fractional Brow-nian sheet following the method of Section 2.1, using rectangles in [0,1]2 instead of intervals in [0,1]. Denote byδWα,β andDWα,β its associated Skorohod integral and Malliavin deriva-tive respecderiva-tively. A consequence of the identity (12) is that the following expression of the Skorohod integral δWα,β

All properties of the anticipating Skorohod integration with respect to Gaussian processes will hold in this case.

Consider now the processes s _→ W_s,tα,β and t _→ W_s,tα,β. These processes are real fractional Brownian motions with the same law as tβ_Wα _{and s}α_Wβ _{respectively and with}

covariances

R1(s1, s2) =t2β

1 2

¡

s2α₁ +s2α_{2 − |}s1−s2|2α ¢

and R2(t1, t2) =s2α

1 2

³

t2β₁ +t2β_{2 − |}t1−t2|2β ´

(13) respectively. Since these processes are fBm’s themselves, we have the Malliavin calculus with respect to them. We denote byδt₁ the Skorohod integral with respect to the fractional Brownian motions_→W_s,tα,β and by δ₂s the Skorohod integral with respect to the fractional Brownian motiont_→W_s,tα,β. We will only use the integration by parts formulas

f(W_s,tα,β)δ₁t(u) =δ₁t³f(W_s,tα,β)u´+t2βf′(W_s,tα,β)_hu,1_[0,s]iHα (14)

and

f(W_s,tα,β)δs₂(u) =δ₂s³f(W_s,tα,β)u´+s2αf′(W_s,tα,β)_hu,1_[0,t]iHβ. (15)

3

The Itˆ

o formula for the fractional Brownian sheet

We will start this section with three technical Lemmas that will be used below, and that rely on the following exponential quadratic growth assumption onf and/or its derivatives:

(H) We say that a functiongonRsatisfies Condition (H) if, for_|x_|large enough, we have

|g(x)_{| ≤}Mexpax2 _{where ais a constant that depends only on α and/or β, and M}

is an arbitrary constant.

Lemma 1 Let s, t _∈ [0,1] and π1 : 0 = s0 < s1 < . . . < sn = s and π2 : 0 = t0 < t1 < . . . < tm = t be two partitions of the intervals [0, s] and [0, t] respectively. Assume f, f′,

and f′′ _{satisfy Condition (H). Then we have the following convergences inL}2_{(Ω)when the}

partition meshes both tend to zero:

1 If (Wα)t∈[0,1] is a fBm with parameter α∈(1₂,1), then for anyf ∈Cb2(R), m−1

X j=0

δ³f(W_tα_j)t_j2α1[tj,tj+1](·)

´

=

m−1 X j=0

Z tj+1

tj

f(W_tα_j)t2α_j dW_uα_→ Z t

0

f(W_uα)u2αdW_uα.

(16)

2 If (W_s,tα,β)s,t∈[0,1] is a fractional Brownian sheet with parameters α, β ∈ (1₂,1), then for

any f _∈C_b2(R),

n−1 X i=0

m−1 X j=0

δ³1[si,si+1]×[tj,tj+1](·)f(W

α,β si,tj)

´

=

n−1 X i=0

m−1 X j=0

Z si+1

si

Z tj+1

tj

f(W_sα,β_i,tj)dW_u,vα,β (17)

→ Z t

0 Z s

0

Proof: We use the notation ∆j = [tj, tj+1]. Concerning point 1, we prove first the

to the process

f(W_uα)u2α1[0,t](u) = m−1

X j=0

f(W_uα)u2α1∆j(u). (20)

The L2-norm of the differenceDbetween the two terms can be estimated as follows

D=E

This follows from the mean value theorem and the fact thatW is almost-surely continuous. Since here we will have u_∈∆j,so that|u−tj| ≤

Therefore, using H¨older’s inequality,

D_≤2E

f′ witha <8−1V ar(Z)−1 whereZ = sup_x∈[0,1]|W_xα_|. Indeed, sinceZ is the supremum of a centered continuous Gaussian process, a classical result of Fernique implies thatE£

expaZ2¤

is finite fora < V ar(Z)−1₂−1_{. Using the inequality (2), the proof of point 1 will be finished}

if we show that the derivative of (19) converges to the derivatives of (20). It holds that

Dsf(Wtαj) =f convergence of the first two terms to zero can be done using the bounds presented above in this proof, using Condition (H) on f′ and f′′. Finally, using the definition of the scalar product in (_Hα)⊗2, including the fact that we can use the inequality 1_[tj,u](s)_≤1∆j(s), we

Condition (H) onf′ _{guarantees that E}h_sup

x∈[0,1]|f′(Wxα)|2 i

is finite, and the other factor converges to 0 with¯

¯π2¯

Concerning point 2, we need first to prove that

goes to zero in L2¡

due to the almost-sure continuity of Wα,β as a centered Gaussian process on [0,1]2. The convergence of the Malliavin derivatives of (21) is no more difficult than the other proofs above; we leave it to the reader. The proof of the lemma is completed. ¤

The next lemma is trivial.

Lemma 2 Let s, t_∈ [0,1], π1_{, π}2 _{as in Lemma 1 and A, B two finite variation processes.}

The next lemma is useful for dealing with certain specific double Skorohod integrals that we will encounter.

Lemma 3 Fors, t_∈[0,1]and for fixed integers n andm, let the partitions π1, π2 be again

as the mesh of π tends to 0 (whenn and m both tend to infinity) to the function adefined

for every(u1, v1),(u2, v2)∈[0,1]2 by:

Proof: We denote s0 =t0 = 0 and ∆i = ∆ni = [si;si+1], ∆j = ∆mj = [tj;tj+1],as

well as Di = Dn_i = [0, si] and Dj = D_jm = [0, tj]. Although the names of these intervals are ambiguous, the names of their indices allow us to identify which variables they refer to, which lightens the notation. To further lighten notation, we allowDj(t) to denote the

indicator function 1Dj(t), and similarly for the other intervals. We only need to show that

the doubly indexed sequence of functions

[an,m((u1, v1),(u2, v2))₋a((u1, v1),(u2, v2))]

·£

an,m¡¡u′1, v′1 ¢

,¡

u′₂, v′₂¢¢ −a¡¡

u′₁, v′₁¢ ,¡

u′₂, v′₂¢¢¤

=

 

n−1 X i=0

m−1 X j=0

Dj(v1) ∆i(u1)Di(u2) ∆j(v2)−a((u1, v1),(u2, v2))  

·  

n−1 X i=0

m−1 X j=0

Dj ¡

v′₁¢

∆i ¡

u′₁¢ Di

¡ u′₂¢

∆j ¡

v′₂¢ −a¡¡

u′₁, v₁′¢ ,¡

u′₂, v₂′¢¢ 

 (25)

defined on¡

[0,1]2_×[0,1]2¢2 _{converges to 0 as m and ntend to infinity in the spaceL}1_(µ)

whereµ is the measure defined by

µ¡

du1du2dv1dv2du1′du′2dv1′dv2′ ¢

=¯¯u₁−u′₁ ¯ ¯

2α−2¯

¯u₂−u′₂ ¯ ¯

2α−2¯ ¯v₁−v₁′

¯ ¯

2β−2¯ ¯v₂−v₂′

¯ ¯

2β−2

du1du2dv1dv2du1′du′2dv′1dv2′.

We first note that, by virtue of the partitions, for every pair of points (u1, v1),(u2, v2) _∈ [0,1]2, except for the countably many points in the partitions π1 and π2, we have con-vergence of an,m((u1, v1) ; (u2, v2)) to a((u1, v1) ; (u2, v2)). In other words, an,m converges

to a Lebesgue-almost everywhere in [0,1]2_×[0,1]2. Thus the quantity in (25) converges in ¡

[0,1]2_×[0,1]2¢2 almost everywhere to 0 with respect to the Lebesgue measure, and also with respect to the measure µ since µ is Lebesgue-absolutely continuous. Since the sequence in (25) is bounded by 1, and since 1 is integrable with respect toµ, by dominated

convergence, the lemma is proved. ¤

We will also need the basic lemma for the convergence of the Skorohod integral.

Lemma 4 Letunbe a sequence of elements inDom(δ)which converges touinL2 ¡

Ω;_Hα,β¢

.

Suppose thatδ(un) converges in L2(Ω) to some square integrable random variable G. Then

u belongs to the domain of δ and δ(u) =G. This result also holds for sequences in_Hα_{, or}

in _Hβ, and their respective Skorohod integrals.

3.1 A decomposition for (W_s,tα,β₎2

Let, from now on, s, t_∈[0,1] and π1 : 0 =s0 < s1 < . . . < sn =s and π2 : 0 = t0 < t1 < . . . < tm =t be two partitions of the intervals [0, s] and [0, t] respectively. Without loss of

Proposition 1 Let ³W_s,tα,β´

s,t∈[0,1] a fractional Brownian sheet with parameters α, β ∈

(1₂,1). Then it holds

where, withaandaπ _{as in Lemma 3, and withδ}(2) _{the double Skorohod integral with respect}

to Wα,β on [0,1]2_×[0,1]2,

Although this result is contained in the Itˆo formula that we prove independently in the next section, we give a detailed self-contained proof of this decomposition for¡

Wα,β¢2_as

a dydactic tool that enables us to introduce some of the essential ingredients and techniques used in proving Itˆo’s formula, including the random field ˜M. This section helps us make the next section more concise and easier to read.

Proof of the proposition: First write Taylor’s formula (W_s,tα,β)2= 2

is equal to 0. Indeed, by H¨older inequalities,

and this goes to 0 when α > 1₂. Therefore it suffices to study the limit of the sum

By Lemma 1 point 2, and Lemma 2, we have the following convergences in L2(Ω)

The term J admits the following decomposition

J = 2

n−1 X i=0

m−1 X j=0

δh³W_sα,β_i,tj+1−W_sα,β_i,tj´1_∆i×[0,tj]i+ 2

n−1 X i=0

m−1 X j=0

h1_[0,si],1∆iiHαh1[0,tj],1∆jiHβ

= 2M(π) + 2

n−1 X i=0

m−1 X j=0

h1[0,si],1∆iiHαh1[0,tj],1∆jiHβ

where

M(π) =

n−1 X i=0

m−1 X j=0

δh³W_sα,β_i,tj+1−W_sα,β_i,tj´1_∆i×[0,tj]i

=δ(2)  

n−1 X i=0

m−1 X j=0

1_∆i×[0,tj](_·)1_[0,si]×∆j(⋆)



=δ(2)(aπ) (30)

whereδ(2) denotes the double Skorohod integral and the processaπ is defined in Lemma 3. As before, we can show that

n−1 X i=0

m−1 X j=0

h1[0,si],1∆iiHαh1[0,tj],1∆jiHβ →|π|→0

1 4s

2α_t2β _inL2 _{. (31)}

Lemma 4 can be used in combination with Lemma 3 to assert immediately that δ(2)_(aπ₎

converges in L2(Ω) to δ(2)(a); indeed the convergence of the deterministic integrands in

¡

Hα,β¢⊗2

is in Lemma 3, and since the corresponding Skorohod integrals are equal to (W_s,tα,β)2 minus other terms that were already proved to converge inL2(Ω), they converge in L2_{(Ω). Note that Lemma 4 is not stated for the convergence of the double Skorohod}

integral, but the simple one; however it can be still applied because a double integral can be written as a simple one in which the integrand is convergent inL2(Ω;_Hα,β)). Combining this convergence and (28), (29), (31), we find the result of the proposition. ¤

3.2 Itˆo formula

In the sequel of this article we will make use of the notationduWu,vα,βto denote the differential

of the f Bm given by u _7→ Wu,vα,β for fixed v, and similarly for dvWu,vα,β. As before, dWu,vα,β

denote the differential of the fractional Brownian sheet with respect to both parameters.

(1₂,1). Also assume that f, f′, f′′, f′′′ and f(iv) satisfy Condition (H). Then it holds

f(W_s,tα,β) =f(0) +

Z t 0

Z s 0

f′(W_u,vα,β)dW_u,vα,β

+ 2αβ Z t

0 Z s

0

f′′(W_u,vα,β)u2α−1v2β−1dudv+

Z t 0

Z s 0

f′′(W_u,vα,β)dM˜u,v

+α Z t

0 Z s

0

f′′′(W_u,vα,β)u2α−1v2βdudvWu,vα,β+β Z t

0 Z s

0

f′′′(W_u,vα,β)u2αv2β−1duWu,vα,βdv

+αβ Z t

0 Z s

0

fiv(W_u,vα,β)u4α−1v4β−1dudv (32)

where, by definition,

Z t 0

Z s 0

f′′(W_u,vα,β)dM˜u,v = lim

|π|→0δ (2)

 

n−1 X i=0

m−1 X j=0

f′′³W_sα,β_i,tj´1_{[0,si]×[tj,tj+1]}(_·)1_{[si,si+1]×[0,tj]}(⋆)

 

where the last limit exists inL2_{(Ω). More specifically, we have the following formula, which}

can be taken as a definition of the integral with respect toM˜ which would otherwise only be

a formal notation:

Z t 0

Z s 0

f′′(W_u,vα,β)dM˜u,v :=δ(2)(N) (33)

where for every (u1, v1),(u2, v2)∈[0,1]2 the function N is defined by:

N((u1, v1) ; (u2, v2)) =f′′ ³

W_uα,β_1,v2´1_[0,s](u1) 1[0,t](v2) 1[0,u1](u2) 1[0,v2](v1). (34)

Proof: Let s, t_∈[0,1] and write Taylor’s formula with fixed t. It holds that

f³W_s,tα,β´=f(0) +

n−1 X i=0

f′³W_sα,β_i,t´ ³W_sα,β_i+1,t−W_sα,β_i,t´

+

n−1 X i=0

where W¯_sα,β_i,t is a point on the segment from W_sα,β_i+1,t toW_sα,β_i,t. First, we can prove that the last term goes to zero in L2(Ω). Indeed, by H¨older’s inequality,

E

where K is a constant obtained by using Condition (H) on f′′,C is a universal constant; this all tends clearly to zero since 2α >1. It remains to study the limit of the term

T =

3.2.1 The estimation of the term A

Note that

³

W_sα,β_i+1,tj+1−W_sα,β_i,tj+1−W_sα,β_i+1,tj +W_sα,β_i,tj´=δ¡

1∆i×∆j(·)

and by the properties of the Skorohod integral we obtain

A=

n−1 X i=0

m−1 X j=0

δhf′³W_sα,β_i,tj+1´1∆i×∆j(·)

i

+

n−1 X i=0

m−1 X j=0

f′′³W_sα,β_i,tj+1´_h1_{[0,si]×[0,tj+1]},1∆i×∆jiHα,β

=A1+A2.

The convergence of A1 follows from Lemma 1 point 2, assuming condition (H) for f′, f′′

and f′′′, yielding

n−1 X i=0

m−1 X j=0

δhf′³W_sα,β

i,tj+1

´

1∆i×∆j(·)

i

→L2_(Ω)

Z t 0

Z s 0

f′(W_u,vα,β)dW_u,vα,β. (35)

Concerning the term A2, we can write

A2 =

n−1 X i=0

m−1 X j=0

f′′³W_sα,β_i,tj+1´_h1[0,si],1∆iiHαh1[0,tj+1],1∆jiHβ

=

n−1 X i=0

m−1 X j=0

f′′³W_sα,β_i,tj+1´1

4

¡

s2α_i+1−s2α_{i −}(si+1−si)2α ¢³

t2β_j+1−t2β_{j −}(tj+1−tj)2β ´

and as above this converges, due to Lemma 2, to

Z t 0

f′′³W_u,vα,β´dRα_udRβ_v =αβ Z t

0

f′′³W_u,vα,β´u2α−1v2β−1dudv (36)

whereRα_u =Rα(u, u) =u2α andRβv =Rβ(v, v) =v2β. due to Lemma 2. 3.2.2 The estimation of the term B

Since

f′³W_sα,β_i,tj+1´₋f′³W_sα,β_i,tj´=f′′³W_s¯α,β_i,tj´ ³W_sα,β_i,tj+1−W_sα,β_i,tj´

whereW_s¯α,β_i,tj is a point located betweenW_sα,β_i,tj andW_sα,β_i,tj+1, the termB becomes

B =X

i,j

f′′³W_s¯α,β_i,tj´ ³W_sα,β_i,tj+1−W_sα,β_i,tj´ ³W_sα,β_{i+1,tj −}W_sα,β_i,tj´

and using the argument as above, assuming Condition (H) for f′′′, we can prove that the limit of B is the same as the limit of the term

B′ =X

i,j

f′′³W_sα,β

i,tj

´ ³ W_sα,β

i,tj+1−W

α,β si,tj

´ ³ W_sα,β

i+1,tj−W

α,β si,tj

It holds that

The second term is similar with the term A2 studied before and we have its convergence to Z t

and by applying the integration by parts several times for the Skorohod integral with respect to the corresponding fractional Brownian motion, we obtain

B1=

For the summandB12we have

By Lemma 1 point 1, assuming condition (H) for f′′′and f(iv), it holds that, for everysi m−1

X j=0

δsi

2 h

f′′′³W_sα,β_i,tj´t2β_j 1∆j(·)

i

→L2_(Ω)

Z t 0

f′′′³W_sα,β_i,v´v2βdvWsα,βi,v

and this fact together with Lemma 2 implies that the first part of B12 goes to the integral

αRs 0

Rt 0f′′′

³ Wu,vα,β

´

u2α−1duv2βdvWu,vα,β. Using Condition (H) onf′′′, since 2α >1 it is easy

to observe that the last part ofB12converges to zero and therefore we have the convergence

B12_→α Z s

0 Z t

0

f′′′³W_u,vα,β´u2α−1duv2βdvW_u,vα,β. (38) In the same way

B13→β Z s

0 Z t

0

f′′′³W_u,vα,β´u2αv2β−1duWu,vα,βdv. (39)

Lemma 2 also gives thatB1,4 converges to

1 4

Z s 0

Z t 0

f(iv)(W_u,vα,β)u2αv2βdRα(u)dRβ(v) =αβ Z s

0 Z t

0

f(iv)(W_u,vα,β)u4α−1v4β−1dudv. (40) Now, note that

B11= X

i,j

δ(2)hf′′³W_sα,β_i,tj´1_[0,si]×∆j(_·)1_∆i×[0,tj](⋆)i

whereδ(2) denotes again the double Skorohod integral, and specifically,

δ(2)hf′′³W_sα,β_i,tj´1[0,si]×∆j(·)1∆i×[0,tj](⋆)

i

=

Z si

0

Z tj+1

tj

Z si+1

si

Z tj

0

f′′(W_sα,β_i,tj)dW_uα,β_1,v1dW_uα,β_2,v2.

We now show

Lemma 5 Using the same partitions as in Lemma 3, define the process

N_(uπ_{1,v1);(u2,v2)}=X

i,j

f′′³W_sα,β_i,tj´1∆i×[0,tj]((u1, v1))1[0,si]×∆j((u2, v2)). (41)

Then assuming condition (H) for f′′ the sequence(Nπ)π converges in L2

³

Ω;¡

Hα,β¢⊗2´

to

the functionN defined in the statement of Theorem 1.

Proof: The proof uses Condition (H) combined with the proof of Lemma 3. We only need to establish the convergence in¡

Hα,β¢⊗2

for fixed randomnessω _∈Ω.Indeed, the convergence in L2³Ω;¡

Hα,β¢⊗2´

for f′′_{, as long as a < 2}−1_{V ar}¡

max_[0,1]2

¯ ¯Wα,β¯

¯ ¢−1

for instance. Now, using the almost-everywhere convergence established in proving Lemma 3, and noticing that we can rewrite

Nπ as

N_(uπ_{1,v1);(u2,v2)}=

 

X i,j

f′′³W_sα,β_i,tj´1∆i(u1)1∆j(v2)

 

·  

X i,j

1_∆i×[0,tj]((u1, v1))1[0,si]×∆j((u2, v2))

 ,

we only need to show that for almost everyω_∈Ω,for fixedu1, v2,the quantity X

i,j

f′′³W_sα,β_i,tj(ω)´1∆i(u1)1∆j(v2)

converges to f′′³_Wα,β u1,v2(ω)

´

. Since Wα,β _{is almost-surely continuous on [0,1]}2_{, this}

con-vergence is trivial. The lemma is proved. ¤

To guarantee that Lemma 4 can be invoked to conclude that the term

δ(2)(Nπ) :=δ(2)  

n−1 X i=0

m−1 X j=0

f′′³W_sα,β_i,tj´1[0,si]×∆j(·)1∆i×[0,tj](⋆)



 (42)

is convergent in L2(Ω) to a Skorohod integral, since we already have the convergence of the integrands in L2³Ω;¡

Hα,β¢⊗2´

from Lemma 5, we only need to guarantee that the Skorohod integrals in (42) converge to a random variable in L2(Ω). However this is trivial since the term in (42) is the difference of all the other terms that have been estimated in this proof, and have been proved to converge in L2(Ω) to f(W_s,tα,β) minus the sum of all terms on the right hand side of (32) except for the one denoted by

Z t 0

Z s 0

f′′³W_u,vα,β´dM˜u,v.

Therefore, the Skorohod integral in (42) converges to the Skorohod integral δ(2)(N), and

(32) is established, so Theorem 1 is proved. ¤

4

Interpretation of the integral with respect to

_M

˜

As we mentioned before in the statement of the Itˆo formula (Theorem 1), the notation ˜M

differential notation for ˜M is the following. We can write

˜

M(s, t) =

Z u1,v1

Z u2,v2

dW_uα,β_1,v1dW_uα,β_2,v21_[0,s](u1) 1[0,t](v2) 1[0,u1](u2) 1[0,v2](v1)

=

Z s u1=0

Z t v2=0

Z u₁ u2=0

Z v₂ v1=0

dW_uα,β_1,v1dW_uα,β_2,v2 (43)

=

Z s u1=0

Z t v2=0

µZ v2

v1=0

dW_uα,β_1,v1

¶ µZ u1

u2=0

dW_uα,β_2,v2 ¶

(44)

=

Z s u1=0

Z t v2=0

du1W

α,β

u1,v2 ·dv2W

α,β

u1,v2 (45)

which shows thatdM˜ (s, t) should be interpreted as

dsWs,tα,β·dtWs,tα,β. (46)

To make the string of equalities above rigorous, the Fubini-type property used to go from (43) to (44) must be justified, and the double stochastic integral in (45) must be properly defined. This can be done without needing to invoke a new Skorohod integration theory, by simply noting that the integral in (45) can be defined as a random variable in the second Gaussian chaos of Wα,β with an appropriate distribution, and checking that it equals ˜M(s, t). We omit these details.

Defining stochastic integration with respect to ˜M is also trivial if the integrands are deterministic. However, in connection with the general Itˆo formula (Theorem 1), we are much more interested in random integrands. A proper theory of general Skorohod integration with respect to ˜M can be given. We will not present this theory here, since it constitutes only a tangent to the Itˆo formula that is the subject of this article, and especially since it is not clear that it brings any more computational information than formulas (33) and (34) which can be taken as a definition of general Skorohod integration against ˜M. What we will do instead is to present briefly a somewhat formal calculation that shows exactly how formula (33) is related to integration against ˜M. The main point in this calculation is that, although the integrands are non-deterministic, the familiar Malliavin-derivative-type corrections that come from pulling random non-time-dependent terms out of Skorohod integrals do not appear in the final formula. We recall the general formula for Skorohod integration with respect to an arbitrary Gaussian field, and a fixed L2 random variableF:

F δ(u) =δ(uF) +_hu;DF_iH

where_h·;_·iH is the inner product in the canonical Hilbert space of the underlying Gaussian field.

We also recall that the stochastic differential of Wα,β can be formally understood as follows, using a standard Brownian sheet W:

dW_u,vα,β=dudv Z u

a=0 Z v

b=0 ∂Kα

∂u (u, a) ∂Kβ

Therefore we have

Now we pull the constant L2(Ω)-random variable f(u, v) into the integral with respect to

dWa,b, modulo a correction term, then use Fubini to pull the Riemann integral with respect

tou′ all the way in (which requires no correction), yielding

I =

Now we perform the same operation onf and the integral with respect to v′_{, yielding}

=dudv ·Z u

a=0 Z v

b=0

Kα(u, a)∂K

β

∂v (v, b)dWa,b ¸

·f(u, v)

·Z u a′₌₀

Z v b′₌₀

∂Kα ∂u

¡ u, a′¢

Kβ¡ v, b′¢

dWa′_,b′ ¸

=f(u, v)

· dv

Z v b=0

∂Kβ

∂v (v, b)dbW α,1/2 u,b

¸ · du

Z u a′₌₀

∂Kα ∂u

¡ u, a′¢

da′W1/2,β a′_,v

¸

=f(u, v)dvWu,vα,β·duWu,vα,β.

Modulo the definition of Skorohod integration with respect todvWu,vα,β·duWu,vα,β, this

calcu-lation justifies the formula

Z t 0

Z s 0

f(u, v)dM˜u,v

:=

Z [0,1]2

Z [0,1]2

dW_uα,β_1,v1dW_uα,β_2,v2f(u1, v2) 1[0,s](u1) 1[0,t](v2) 1[0,u1](u2) 1[0,v2](v1)

=

Z t 0

Z s 0

f(u, v)dvW_u,vα,β_·duW_u,vα,β,

which can be summarized by

dMu,v˜ =dvW_u,vα,β_·duW_u,vα,β

as announced in (46). ¤

5

Application to the local time of the fractional Brownian

sheet

As an application of the Itˆo formula, we will establish in this section the existence and the stochastic integral representation of the local time of the fractional Brownian sheet.

5.1 Known results and motivation

We start with a short summary of the results on the local times for one and multiparameter fractional Brownian motion.

• The local timeλa_t of the one parameter fractional Brownian motionBH was introduced in [5] as the density of the occupation measure Γ_→Rt

01Γ(BHs )ds. The author proved

that λa_t has a jointly continuous version in the variables a and t. Moreover λa_t has H¨older-continuous paths of order δ < 1₋H in time and of order γ < 1_2H−H in the space variable a provided that H _≥ 1₃. As a density of an occupation measure, the local time is increasing in t and the measureLa(dt) is concentrated on the level set

•Another version of the local time of the fractional Brownian motion was introduced in [8] as the density of the occupation measure mt(Γ) = 2HRt

0 1Γ(BsH)s2H−1ds. If H ≥ 13,

this local time satisfy the Tanaka formula

|B_tH ₋a_|=_{| −}a_|+

Z t 0

sign(Ba_s)dB_sH +La_t

where the stochastic integral is defined in the Skorohod sense. This formula was re-cently extended by [7] toH_∈(0,1). The regularity properties ofλa_t can be transferred toLa_t by integrating by parts.

• In [12] the local time of the d-dimensional fractional sheet with N-parameters was studied. This local time can be formally defined as La_{t =} R

[0,t]δa(Ws )H ds, where

t = (t1, . . . , tN) ∈ TN , H = (H1, . . . , HN) and δa denotes the Dirac function. The

smoothness of the local time in the Sobolev-Watanabe spaces and its asymptotic be-havior were studied.

• Recently, in [18], using the techniques of the Fourier analysis the authors proved the existence of the local time of the N-parameters d-dimensional fractional Brownian sheet satisfying the occupation density formula

Z A

f³WH_t´dt =

Z Rd

f(x)L(x, A)dx

for any Borel setA_⊂RN and for any measurable functionf :Rd_→R. The existence of a jointly continuous version of the local time is proved. Obviously, this local time coincides with the one used in [12].

In this section we define the local timeLa

s,t,a∈R of the fractional Brownian sheet ³

W_s,tα,β´

s,t∈[0,1] as the density of the occupation measure ms,t(Γ) =αβ

Z t 0

Z s 0

1Γ(Wu,vα,β)u4α−1v4β−1dudv. (47)

Note that in the case α=β = 1₂ this local time coincide with the local time introduced in [13] and [16] for the standard Wiener process with two parameters.

Remark 1 Using the integration by parts we can transfer the joint continuity of the local

time defined in [18] to La

s,t. The details are left to the reader.

5.2 Stochastic representation of local time

Let

pε(x) =

1

√

2πεe

be the heat kernel with varianceε >0. The following results gives an approximation inL2

of the local time of the fractional Brownian sheet. It can be deduced from [12], where the authors used the Wiener chaos decomposition of the local time.

Proposition 2 Let a_∈R ands, t_∈[0,1] and define

La,ε_s,t =αβ Z t

0 Z s

0

pε(Wu,vα,β)u4α−1v4β−1dudv.

Then the random variable La,ε_s,t converges toLa_s,t in L2(Ω) as εtends to zero.

The next result gives the stochastic integral representation of the local time.

Theorem 2 For any a_∈R and s, t_∈[0,1], we have

La_s,t = 1 6

¯ ¯ ¯W

α,β s,t −a

¯ ¯ ¯

³

W_s,tα,β₋a´2₋ 1

2

Z t 0

Z s 0

¯ ¯ ¯W

α,β u,v −a

¯ ¯ ¯

³

W_u,vα,β₋a´dW_u,vα,β

−2αβ Z t

0 Z s

0 ¯ ¯ ¯W

α,β u,v −a

¯ ¯ ¯u

2α−1_v2β−1_dudv

− Z t

0 Z s

0 ¯ ¯ ¯W

α,β u,v −a

¯ ¯ ¯dM˜u,v −α

Z t 0

Z s 0

sign(W_u,vα,β₋a)u2α−1v2βdvWu,vα,βdu

−β Z t

0 Z s

0

sign(W_u,vα,β₋a)u2αv2β−1duWu,vα,βdv.

Proof: Let us introduce the functions

gIV_ε (x) =pε(x), g ′′′

ε (x) = 2 Z x

0

pε(y)dy−1,

g_ε′′(x) =

Z x 0

g_ε′′′(y)dy, g_ε′(x) =

Z x 0

g_ε′′(y)dy, gε(x) = Z x

0

g_ε′(y)dy.

Clearly, asεtends to zero, we have

g_ε′′′(x)_→g′′′(x) =sign(x), g_ε′′(x)_→g′′(x) =_|x_|,

g_ε′(x)_→g′(x) = 1

2x|x|, gε(x)→g(x) = 1 6x

Let’s write Itˆo’s formula for the function gε(x−a), where ais a real number. We get

We will decompose the proof into several steps.

Step 1. Given the existence of moments of all orders for the supremum of continuous

Gaussian fields, dominated convergence implies the following convergences hold in L2_(Ω): gε(Ws,tα,β−a)→

and by Proposition 2,

αβ

Step 2. We will show that the integrand of the stochastic integral with respect to the

fractional Brownian sheet is convergent. More precisely, consider the process

g′_ε(W_u,vα,β)1_[0,t]×[0,s](u, v) and let us show its convergence inL2¡

which goes to zero asε_→0.

Step 3. Lemma 5 together with the proof of Theorem 1 shows that the limit in L2(Ω)

lim |π|→0δ

(2)X i,j

h

f′′³W_sα,β_i,tj´1[0,si]×∆j(·)1∆i×[0,tj](⋆)

i

exists for everyf satisfying Condition (H). Consequently, for everyε >0, denote byAεthe following random variable inL2(Ω):

Aε= lim

|π|→0δ (2)X

i,j h

g′′_ε³W_sα,β_i,tj´1[0,si]×∆j(·)1∆i×[0,tj](⋆)

i .

Since g′′

ε converges to g(x) = 12|x|x, we obtain by Lemma 4 (with the observation made

at the end of the proof of Proposition 1) the convergence in L2(Ω) of Aε, as ε goes to

zero, to the random variable lim_|π|→0δ(2)P i,j

h

g³W_sα,β_i,tj´1_[0,si]×∆j(_·)1_∆i×[0,tj](⋆)iwhich is inL2(Ω).

Step 4. We consider now the term

β Z t

0 Z s

0

g_ε′′′(W_u,vα,β₋a)u2αv2β−1dvduW_u,vα,β

and we will prove its convergence to

β Z t

0 Z s

0

sign(W_u,vα,β₋a)u2αv2β−1dvduW_u,vα,β.

An argument from the one-parameter case can be used to conclude this fact. The proof of this step follows from the next lemma and the dominated convergence theorem.

Lemma 6 Consider Bα _{a one-dimensional fBm with α ∈ (}1

2,1). Then, for every s ∈ T,

Rs

0 gε′′′(Buα−a)u2αdWuα converges in L2(Ω) to Rs

0 sign(Buα−a)u2αdWuα.

Proof: Let us write Itˆo’s formula withhε(t, x) =t2αg′′ε(x) (this is a straightforward

application of Theorem 1 of [1]). We get

s2αg′′_ε(B_sα₋a) = 2α Z s

0

g_ε′′(B_uα₋a)u2α−1du+

Z s 0

g_ε′′′(B_uα₋a)u2αdB_uα

+ 2α Z s

0

pε(B_uα₋a)u4α−1du

The following convergences in L2(Ω) can be proved exactly as in [8]:

2α Z s

0

g_ε′′(B_uα₋a)u2α−1du_→2α Z s

0 |

B_uα₋a_|u2α−1du

2α Z s

0

pε(Buα−a)u4α−1du→ Z s

0

La_uu2αdu

(here La

u is the local time of the one-parameter fBm defined in [8] and presented at the

beginning of this section), and

1_[0,s](u)g_ε′′′(B_uα₋a)u4α−1 _→1_[0,s](u)sign(Bα_{u −}a)u2α,

the last convergence being in L2(Ω;_H). That gives the conclusion using Lemma 4. ¤

Lemma 4 again and Steps 1 to 4 above finish the proof of the theorem. ¤

Remark 2 Using standard arguments, additional versions of the stochastic integral repre-sentation of the local time of the fractional Brownian sheet can be obtained by using, instead

of the functions g, g′, g′′ and g′′′ appearing in the proof of Theorem 2, the functionsj, j′, j′′

and j′′′, where

j(x) = 1

6[(x−a)

+_]3_{, j}′

(x) = 1

2[(x−a)

+_]2

and

j′′(x) = (x₋a)+ , j′′′(x) = 1_(a,∞)(x).

More precisely, we have

1 2L

a

s,t =j(Ws,tα,β)− Z t

0 Z s

0

j′³W_u,vα,β₋a´dW_u,vα,β

−2αβ Z t

0 Z s

0

j′′³W_u,vα,β₋a´u2α−1v2β−1dudv

− Z t

0 Z s

0

j′′³W_u,vα,β₋a´dMu,v˜

−α Z t

0 Z s

0

j′′′³W_u,vα,β₋a´u2α−1v2βdvWu,vα,βdu

−β Z t

0 Z s

0

j′′′³W_u,vα,β₋a´u2αv2β−1duWu,vα,βdv.

Negative part functions can be also used to express the local time.

Our last result is an occupation time formula for the local time of the fractional Brownian sheet, for which the previous remark is useful.

Proposition 3 For any Borel function f :R_→ R and and for any a_∈ R, s, t_∈[0,1], it holds that

2αβ Z t

0 Z s

0

f(W_u,vα,β)u4α−1v4β−1dudv =

Z R

Proof: We use the same idea as in [16]. Let us introduce the function hIV_x (y) = 1_(0,x)(y) and its anti-derivatives. Notice that

h′′′_x(y) =

Z x 0

1_(y>x′₎dx′, h ′′ x(y) =

Z x 0

(y₋x′)+dx′,

and

h′_x(y) =

Z x 0

1

2[(y−x

′₎+_]2_dx′_{, hx(y) =}Z x

0

1 6[(y−x

′₎+_]3_dx′_.

On the one hand, we can write Itˆo’s formula forhx(y) (although the fourth derivative

is not continuous, is it not difficult to see that Itˆo’s formula still holds, by approximating

hx by smooth functions) and we get

αβ Z t

0 Z s

0

1(0,x)(Wu,vα,β)u4α−1v4β−1dudv

=hx(Ws,tα,β)−hx(0)− Z t

0 Z s

0

h′_x(W_u,vα,β)dW_u,vα,β

−2αβ Z t

0 Z s

0

h′′_x(W_u,vα,β)u2α−1v2β−1dudv₋ Z t

0 Z s

0

h′′_x(W_u,vα,β)dM˜u,v

−α Z t

0 Z s

0

h′′′_x(W_u,vα,β)u2α−1v2βdudvW_u,vα,β₋β Z t

0 Z s

0

h′′′_x(W_u,vα,β)u2αv2β−1duW_u,vα,βdv.

On the third hand, using the integral expression of the local timeLy_s,t given in Remark 2, multiplying both sides by 1_(0,x)(y), integrating over the real line and changing the order of integration (see exercise 3.2.8 of [17] for the Fubini anticipating theorem), we will obtain that

1 2

Z R

1_(0,x)(y)Ly_s,tdy=αβ Z t

0 Z s

0

1_(0,x)(W_u,vα,β)u4α−1v4β−1dudv.

The formula (48) is proved forf(y) = 1_(0,x)(y). A monotone class argument is sufficient to

conclude the proof. ¤

5.3 Relation with the variation of M˜

Whenα=β= 1₂ the process ˜M is a martingale and in this case (see [16]) the formula (48) can be written as

1 2

Z t 0

Z s 0

f(W_u,vα,β)d_hM˜_iu,v =

Z R

f(a)La_s,tda. (49)

This can be seen immediately by formally taking the square of the differential dM˜u,v = duM˜u,v·dvM˜u,v which yields

³ dM˜u,v

´2

=uvdudv.

As with one-parameter fBm, there is a specific non-quadratic variation of ˜M that will be non-zero and non-infinite. One can easily prove that, for smallh, k >0, andp, q >1

Eh³W_u+h,vα,β ₋W_u,vα,β´p³W_u,v+kα,β ₋W_u,vα,β´qi=vβphαpuαqkβq+o³max³hαp, kβq´´. (50) In order for the increment

dM˜u,v =dvWu,vα,β·duWu,vα,β= ³

W_u+du,vα,β ₋W_u,vα,β´p³W_u,v+dvα,β ₋W_u,vα,β´

to yield the differential of a non-trivial bounded variation process, one must choose the powers p and q above in order to get the product dudv. Therefore we define the (p, q )-variation of ˜M by the formula

dDM˜E(p,q) u,v =d

D

W_·α,β_,v E(p) u d

D

W_u,α,β_· E(q)

v (51)

where for any number r > 1, the r-variation of a one-parameter process X is defined as usual using partitionsπ =_{t1,_{· · ·}, tn_}of [0, t] by

hX_i(r)(t) = lim |π|→0

n X i=1

¯

¯X_ti−X_ti−1 ¯ ¯ r

. (52)

Formula (50) proves that DM˜E(α −1_,β−1₎

u,v exists and is non-zero, and in fact

dDM˜E(α −1_,β−1₎

u,v =u

α/β_vβ/α_dudv.

Despite the somewhat formal nature of the above development, the definition of

D

˜

ME(α −1_,β−1₎

u,v can be made rigorous because of the tensor-type nature of ˜M, and the last

formula can be established rigorously by only referring to the definition of ˜M as a double Skorohod integral in the statement of Proposition 1 or Theorem 1. By comparing with formula (48) in Proposition 3, we can state the following, which answers the issue of the validity of formula (49) negatively, leaving a rigorous proof, and a rigorous definition of

D

˜

ME, to the reader.

Proposition 4 The (p, q)-variation of the fractional Brownian sheet (u, v) _7→ DM˜E(p,q) u,v ,

defined in equations (51) and (52), is a non-trivial bounded-variation process on [0,1]2 if

and only if p= 1/α and q= 1/β. In that case

D

˜

ME(α −1_,β−1₎

s,t =

Z s 0

Z t 0

dudvuα/βvβ/α.

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