White noise approach to multiparameter stochastic integration

Abstract

In this paper we will set up the Hida theory of generalized Wiener functionals using *(d), the space of tempered distributions on d, and apply the theory to multiparameter stochastic integration. With the partial ordering on +d: (s1, ..., sd) , [xi] [epsilon] I*(Td) is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L2(*(d)) = [Sigma]n = 0[infinity][circle plus operator]Kn, where Kn is the space of n-tuple Wiener integrals. As in the one-dimensional case, there are the continuous inclusions (L2)+ [subset of] L2(I*(Rd)) [subset of] (L2- and (L2)- is considered the space of generalized Wiener functionals. We prove that the multidimensional Ito stochastic integral is a special case of an element of (L2)-. For d = 2 the Ito integral is not sufficient for representing elements of L2(*(2)). We show that the other stochastic integral involved can also be realized in the Hida setting. For F[set membership, variant]*() we will define F(W(s, t), x) as an element of (L2)- and obtain a generalized Ito formula.Hida theory multiparameter stochastic integral Ito formula