Multiple Wiener-ITÔ Integral Processes with Sample Paths in Banach Function Spaces

Abstract

Fix an integer m ≥ 1. Consider a measurable stochastic process I _m which has a representation in terms of m-th multiple Wiener-Itô integrals I _m(ht), t ∈ T, of a time dependent kernel h. Namely, let I _m = {I _m(h _t);t ∈ T} with

$$I_m(h_t)=\int \cdots \int h_t(x_1,\cdots,x_m)G(dx_1)\cdots G(dx_m),\;\;\;\;\;t \in T,$$

where h = h _t = {h(t, ·); t ∈ T} is a family of square summable functions defined on the product measure space (X ^m, v ^m) and G is a Gaussian random measure with the control measure v. The problem we study in this paper is as follows: given a measurable function h on T × X ^m and given a Banach function space F(T, μ) of measurable functions on a separable σ-finite measure space (T,μ), when does

$$I_m \in F(T,\mu)\;\;\;\;\;\;almost\;surely(a.s.)?\;\;\;\;\;\;\;(1.1)$$

If m = 1 then I ₁ is a Gaussian process with the covariance function

$$T\times T \ni (t,s) \mapsto \int_X h_t(x) h_s(x)\nu (dx)$$

.