White Noise Approach to Multiparameter Stochastic Integration.

Abstract

In this dissertation we will set up the Hida theory of generalized Brownian functionals, or white noise analysis, on ${\cal L}\sp{\*}(\IR\sp{\rm d})$, the space of tempered distributions, and apply it to multiparameter stochastic integration. With the partial ordering on $\IR\sbsp{+}{\rm d}$: (s$\sb1$, dots,s$\sb{\rm d}$) $<$ (t$\sb1$, dots,t$\sb{\rm d}$) if s$\sb{\rm i}$ $<$ t$\sb{\rm i}$, 1 $\leq$ i $\leq$ d, the Wiener process W((t$\sb1$, dots,t$\sb{\rm d}$),x) = $\langle$x,1$\sb{\rm \lbrack 0,t\sb1)\times\cdots \times\lbrack 0,t\sb{d})}\rangle$, x $\in {\cal L}\sp{\*}(\IR\sp{\rm d})$ is a generalization of a Brownian motion and there is the Wiener-Ito decomposition: L$\sp2({\cal L}\sp{\*}(\IR\sp{\rm d}))$ = $\sum\sbsp{\rm n = 0}{\infty}\times$K$\sb{\rm n}$, where K$\sb{\rm n}$ is the space of n-tuple Weiner integrals. As in the one-dimensional case, there are the continuous inclusions (L$\sp2)\sp+ \subset$ L$\sp2({\cal L}\sp{\*}(\IR\sp{\rm d})) \subset$ (L$\sp2)\sp-$, and (L$\sp2)\sp-$ is considered the space of generalized Wiener functionals. We will define the differentiation operator $\partial\sb{\rm (t\sb1,\dots,t\sb{d}})$ and its adjoint $\partial\sp{\*}\sb{\rm (t\sb1,\dots,t\sb{d})}$ and give some properties. We prove that the multidimensional time Ito stochastic integral is a special case of a white noise integral and give conditions for its existence. For d = 2 the Ito integral is not sufficient for representing elements of L$\sp2({\cal L}\sp{\*}(\IR\sp2))$. We show that the other integral involved can also be realized in the white noise setting. For F $\in {\cal L}\sp{\*}(\IR\sp{\rm d}$) we will then define F(W((s,t),x) as an element of (L$\sp2)\sp-$ and obtain a generalized Ito formula.