## LSU Historical Dissertations and Theses

1992

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Hui-Hsiung Kuo

#### Abstract

Let $W(t,\omega)$ be a Brownian motion on an abstract Wiener space (i,H,B) corresponding to the canonical normal distribution on H. The well known theorem of Girsanov is proved for such a process with the perturbing term taking values in the Hilbert space H. Consider the stochastic integral equation $\xi(t) = x+\int\sbsp{0}{t}A(s,\xi(s))dW(s)+\int\sbsp{0}{t}\sigma(s,\xi(s))ds$ where x is in $B,\ A(\cdot,\cdot)$ and $\sigma(\cdot,\cdot)$ are bounded continuous functions from $\lbrack 0,\infty)\times B$ to $I+L\sb2(H)$ and H respectively. Here $L\sb2(H)$ denotes the collection of Hilbert Schmidt operators on H. Furthermore, suppose for every $s\ge 0$ the restriction of $A(s,\cdot)$ to H is invertible and $A(s,\cdot)$ and $\sigma(s,\cdot)$ are both Frechet differentiable in the directions of H with bounded derivatives. Under suitable conditions, it is proved that for each $t\ge 0,$ the measure generated by the solution $\xi(t)$ of the above stochastic integral equation is differentiable in the directions of H in the sense of Fomin. By adding more conditions on A and $\sigma,$ it is shown that the transition probability associated with the solution of the stochastic differential equation $d\xi(t)=A(t,\xi(t))dW(t)+\sigma(t,\xi(t))dt$ satisfies the infinite dimensional Kolmogorov's forward equation in the distribution sense.

77

COinS