PRINCIPLE FOR MULTIPLE STOCHASTIC ...

Bol. Soc. Mat. Mexicana (3) Vol. 8, 2002

PRINCIPLE FOR MULTIPLE STOCHASTIC FRACTIONAL INTEGRALS

´ VICTOR PEREZ-ABREU AND CONSTANTIN TUDOR

Abstract. Multiple integrals with respect to a fractional Brownian motion are defined explicitely in terms of multiple Wiener integrals and fractional integrals and derivatives of deterministic functions of several variables. As applications we introduce the chaos form of the fractional stochastic integral and we derive the strong law of large numbers for the fractional Brownian motion.

1. Introduction Fractional Brownian motion (fBm) is playing an increasing role in modeling long-range dependence phenomena in many fields such as telecommunication network, economics, hydrology and biology. From the theoretical point of view a fBm B = {BtH }t∈T is a Gaussian process which is a suitable generalization of the Brownian motion W = {Wt }t∈T but exhibiting long-range dependence. Since the pioneering work by Itô [15], multiple integrals with respect to a Brownian motion and their corresponding chaos expansions have been studied by many authors, being standard tools for dealing with nonlinear functionals of W as well as with anticipating stochastic calculus for W. This is true for both the Wiener-Itô and the Stratonovich multiple integrals (see for example the book by Nualart [23]). It seems then natural to study multiple fractional integrals and in particular chaos expansions for a fractional Brownian motion. This has recently been started by Dasgupta and Kallianpur [4], [5] and Duncan, Hu and Pasik-Duncan [8] for the fBm of Hurst parameter H > 12 , and for general Gaussian process twenty years ago by Huang and Cambanis [14]. The techniques used in these papers involve Wick product and reproducing kernel Hilbert space theory. In this paper, our starting point of view is the representation H Bt = K(t, s)dWs of the fractional Brownian motion as a (single) Wiener integral of a nonrandom fractional kernel K(t, s), with respect to the Wiener process W defined in the same probability space of the fractional Brownian motion B H (see Barton ¨ and Poor [2], Decreusefond and Ustunel [7], Norros et al. [22] and Pipiras and Taqqu [24], [25]). Using this representation we are able to define in an explicit manner multiple fractional stochastic integrals in terms of the classical Wiener-Itô or Stratonovich multiple integrals of the Brownian motion 2000 Mathematics Subject Classification: 60H05. Keywords and phrases: fractional Brownian motion, multiple fractional integrals, chaos decomposition.

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and fractional integrals and derivatives for deterministic functions of several variables. By using this transfer principle, we show that our easily defined multiple fractional stochastic integrals coincide with those defined by Dasgupta and Kallianpur [4], [5] and Duncan, Hu and Pasik-Duncan [8] by more complicated manners. With this approach it happens that properties like multiple product formula, independence of multiple integrals, Hu-Meyer formula, large deviations can be easily transferred from the corresponding to the Wiener process to to multiple fractional integrals. We also introduce the chaos form of the fractional stochastic integral or the so-called divergence operator. The paper is organized as follows. Section 2 reviews several facts on deterministic fractional integrals and the representation of fractional Brownian motion in terms of a standard Brownian motion. Sections 3 is devoted to multiple fractional integrals and the chaos form of the fractional stochastic integral and as an application we derive the chaos decomposition for the solution of affine fractional equations. Finally, Section 4 studies multiple Stratonovich fractional integrals, including the fractional Hu-Meyer formula and an extension of the strong law of large numbers for the fBm (see Dasgupta and Kallianpur [5], Gladyshev [10] for the case of special kernels). After the paper was submitted for publication, the authors learned from the referees about the recent manuscripts by Elliot and van der Noek [9] and Hu [13] in which similar ideas are used independently (see also Kleptsyna, Le Breton and Roubaud [18] for the special case n = 1). 2. Preliminaries We fix H ∈ (0, 1), 0 < t0 < ∞ and 0 < α < 1. We put T = [0, t0 ] or T = R . For a function f : T n → R we define the Liouville fractional integrals α,n (2.1) It0 − f (x1 , ..., xn ) = 1 n Γ(α)

(2.2) 1 n Γ(α)

t0

t0

... x1

xn

∞

f (t1 , ..., tn ) 1−α dt1 ...dtn , xi ∈ [0, t0 ] , j=1 tj − xj α,n I− f (x1 , ..., xn ) = n

∞

... x1

xn

f (t1 , ..., tn ) 1−α dt1 ...dtn , xi ∈ R. − x t j j j=1

n

The following basic properties of fractional integrals of deterministic functions can be found in Samko, Kilbas and Marichev ([26, Ch 5. Sec. 24]). Theorem (2.3). (a) The Liouville fractional integral Itα,n f (x1 , ..., xn ) 0− α,n (resp. I− f (x1 , ..., xn )) is well defined for almost all x1 , ..., xn if f ∈ L1 [0, t0 ]n (resp. f ∈ Lp (R n ), 1 ≤ p < α1 ). p (b) If 1 < p < α1 , q = 1−αp , then the operators Itα,n : Lp [0, t0 ]n → 0− α,n Lq [0, t0 ]n , I− : Lp (R n ) → Lq (R n ), are bounded and one to one. ⊗n ⊗n α,1 α,n α,1 (c) Itα,n = I , I = I . − − t0 − 0−

PRINCIPLE FOR MULTIPLE STOCHASTIC FRACTIONAL INTEGRALS

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2 n 2 α,n L (R ) then the function ϕ ∈ (d) If f ∈ ITα,n L [0, t0 ]n (resp. f ∈ I− − α,n α,n n 2 n 2 ϕ = f ) is unique L [0, t0 ] (resp. ϕ ∈ L (R )) such that It0 − ϕ = f (resp. I− α,n and we denote it by Dtα,n f (resp. D f ). − − 0 Define

12

2HΓ( 32 − H)

. Γ(H + 12 )Γ(2 − 2H) Next we fix a normalized fBm BtH t∈T with Hurst parameter H, defined on a probability space (Ω, F, P ). We assume that F = B BtH : t ∈ T . Recall that B H is a continuous Gaussian process with: (i) B0H = 0. (ii) For every s, t ∈ T, 1 2H 2H 2H E(BsH BtH ) = |t| + |s| − |t − s| . 2 cH =

(2.4)

(2.5). It is known that there exists a standard Brownian motion Remark WtH t∈T defined on the same probability space (Ω, F, P ) such that: (j) BtH t∈T and WtH t∈T generate the same filtration. (jj) (Representation formula). For each t ∈ T , H KH,T (t, s)dWsH , (2.6) Bt = T

(2.7)

H− 1 H− 1 KH,R (t, s) = cH (t − s)+ 2 − (−s)+ 2 ,

1 KH,[0,t0 ] (t, s) = cH (t − s)H− 2 + t s 12 −H 1 H− 32 −H 1− du 1[0,t) (s) = (u − s) 2 u s

1 t 1 1 , − H, H + , 1 − )1[0,t) (s) , 2 2 2 s where F is the Gauss hypergeometric function. If H > 12 , 1 1 1 H− 1 s 2 −H It0 − 2 xH− 2 1[0,t] (s) , (2.9) KH,[0,t0 ] (t, s) = cH Γ H + 2 1 H− 12 I− 1[0,t) (s) , (2.10) KH,R (t, s) = cH Γ H + 2 (2.8)

1

cH (t − s)H− 2 F (H −

and if H < 12 , (2.11)

(2.12)

1 KH,[0,t0 ] (t, s) = cH Γ H + 2

1

−H

s 2 −H Dt20 − 1

1 xH− 2 1[0,t) (s) ,

1 12 −H D− 1[0,t) (s) , KH,R (t, s) = cH Γ H + 2


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For information on the representation formula (2.6) we refer to [2], [7], [22], [24] and [25]. 3. Multiple fractional integrals (A) Case

1 2