White noise approach to multiparameter stochastic integration

Abstract

In this paper we will set up the Hida theory of generalized Wiener functionals using S ∗( R d ), the space of tempered distributions on R d , and apply the theory to multiparameter stochastic integration. With the partial ordering on R + d : ( s 1, …, s d ) < ( t 1, …, t d ) if s i < t i , 1 ≤ i ≤ d, the Wiener process W((t 1 …, t d), x) = 〈x, 1 [0,t 1)x … x[0,t d) 〉, ξ ϵ I ∗(T d) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L 2( S ∗( R d )) = Σ n = 0 ∞⊕ K n , where K n is the space of n-tuple Wiener integrals. As in the one-dimensional case, there are the continuous inclusions (L 2) + ⊂ L 2(I ∗(R d)) ⊂ (L 2− and ( L 2) − is considered the space of generalized Wiener functionals. We prove that the multidimensional Ito stochastic integral is a special case of an element of ( L 2) −. For d = 2 the Ito integral is not sufficient for representing elements of L 2( S ∗( R 2)). We show that the other stochastic integral involved can also be realized in the Hida setting. For F∈ S ∗( R ) we will define F( W( s, t), x) as an element of ( L 2) − and obtain a generalized Ito formula.