Henstock's multiple wiener integral and henstock's version of hu-meyer theorem

K e y w o r d s M u l t i p l e Wiener integral, Hu-Meyer theorem, Henstock integral. 1. I N T R O D U C T I O N Classically, it is well known that the Riemann approach cannot be used to define stochastic integrals. However, it has been proved that the generalized Riemann approach (with nonuniform meshes), also called the Henstock's approach, can be used to study stochastic integrals. See details in [1-5]. The generalized Riemann approach has successfully been used to give an equivalent definition of the classical stochastic integral. While the classical stochastic integral is defined by' a nonexplicit L 2 procedure, the Riemann approach is well known for its explicitness and directness in defining the stochastic integral. The classical multiple Wiener integral was first studied by Wiener in 1938, see [6]. In the same fashion as the definition of one-dimension stochastic integrals, the multiple Wiener integral was later defined by It6 using the generalized L 2 procedure (see [7] for the detail of the procedure). The generalized Riemann approach has been extended to the study of the multiple Wiener integral, and can be similarly used to give an explicit definition of the classical Wiener integral, see [4]. The classical Wiener integral concerns itself with the integration over the nondiagonal part of the n-dimensional Euclidean space by letting the integrand vanish on the diagonal part. In this paper, we shall apply the generalized Riemann approach to the integration of deterministic functions over both the diagonM and the nondiagonal part of the n-dimensional space. We shall derive the classical Hu-Meyer theorem using our approach. The classical treatment of the Hu-Meyer theorem can be found in [8]. 0895-7177/05/$ see front matter (~) 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2004.03.008 Typeset by fl,.MS-TEX 140 T.-L. TOH AND T.-S. CHEW 2. S E T T I N G A N D D E F I N I T I O N O F T H E I N T E G R A L In this section, we shall define the setting and the multiple stochastic integral using Riemann approach. Let T -[a, b] C [0, 0o) and T m = [a, b] m. In this paper, we shall bold the letter to denote intervals in Tm. For example, I C T m is an interval in T m, while I = f l u 1 I~, where each I / i s a left-open interval in T. DEFINITION 2.1. Let (~, P) be a probability space and W = {Wt(w) : t C [a, b]} be a family of random variables on (t~, P). Then, W is said to be a canonicM Brownian motion i f it satisties the folIowing properties, 1. it has normal increments, that is, Wt W~ has a normal distribution with mean 0 and variance t s, for all t > s (which naturally implies that Wt has a normal distribution with mean 0 and variance t); 2. it has independent increments, that is, Wt Ws is independent of its past, that is, W~, a < u < s < t; and 3. its sample paths are continuous, i.e., for each w E f~, Wt(w) as a function o f t is continuous on [a, b]. DEFINITION 2.2. Let 5 be a positive function defined on T m, x = (~1,~2,... ,~-~) E T m and I = rLm=l I~ be an interval of Tm. An interval-point pair (I, x) is said to be &fine; i f Ik C [~k -5((), ~k + 5(~)], for each k = 1, 2, 3 , . . . , m. Note that ~k may or may not be in Ik for each k = 1 , 2 , 3 , . . . , m . A finite collection D of interval-point pairs {(I(i),~(0) : i = 1, 2, 3 , . . . ,n} is said to be a &fine; division of T m if (i) I(i), i = 1, 2, 3 , . . . , n, are disjoint left-open intervals of T; (ii) Ui~l I(i) = (a, b] m. In this paper, a division D may be simply denoted by D = {(I(~),x(0)} or D = {(I ,x)}. We remark that for any given positive function 5 on T m, a &fine; division of T m exists. This can be proved directly by using continued bisection. NOTATION. It could be seen that T m consists of two parts, namely, the diagonal part of T "~, ~D ~-{(Xl, . . . ,Xra) E Tin: xi = x j , for some i ¢ j } and 13 c = { (X l , . . . , xm) E Tin: xi ~ xy, for any i ~ j } , which is the nondiagonal part of T rn. The nondiagonal set :D c plays the basic role in the construction of the multiple It&Wiener; integral, see [4]. The nondiagonal set can be decomposed to m! open connected sets in T "~. For each ~r C Sin, the group of all permutations of m objects, we define CTr ~--{ ( X l , X 2 , X 3 , . . . , X r n ) E T m : Xzr(1) < Xzr(2) < XTr(3) < " '" <( XTr(m)}, and there are m! such sets. Each of these sets is said to be contiguous to the diagonal :D. For any interval I = (u, v] C I~ and k C ~, let W ( I ) and W k ( I ) denote W (I) = W~ W, and W k (I) = (W~ W , ) k , respectively. Let f : T m ~ R be a real-valued function and D = {(I(~),x(~))} a &fine; division of Tm. Then, S ( f , 5, D) denotes the Riemann sum, S ( f , d , D ) = E f (x (i)) W ( I ( i ) ) , where m j=l if I (i) FI,~ r(i) (i) = l l j = l " j and each I 9 is a left-open interval of T. Let L2(f~) denote the space of all square integrable functions of fL Henstock's Multiple Wiener Integral 141 DEFINITION 2.3. A function f : T "~ --~ ]~ is said to be multiple Wiener integrable to M ( f ) C L2(~) on T m if for every e > O, there exists a positive function 5, such that E ( [ S ( f , 5, D ) M ( f ) I 2) 0, there exists 5(x) > 0, such that E ( I S ( f , 5, D ) M ( I ) I 2) < s, whenever D = {(If0, x(~)) : i = 1, 2, 3 . . . . , n} is a standard 5-fine division of T ' L REMARK. From standard properties of Brownian motion, we know that (A) if Ii = (u~, v~] and Ij = (uj, vii are disjoint, then