An evaluation formula for a generalized conditional expectation with translation theorems over paths

J. Korean Math. Soc. Published online July 17, 2019

Dong Hyun Cho
Kyonggi University

Abstract : Let $C[0,T]$ denote an analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. For a partition $0=t_0<t_1<\cdots<t_n<t_{n+1}=T$ of $[0,T]$, define $X_n:C[0,T]\to\mathbb R^{n+1}$ by $X_n(x)=(x(t_0),x(t_1),\ldots,x(t_n))$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $X_n$ which has a drift and does not contain the present position of paths. As applications of the formula with $X_n$, we evaluate the Radon-Nikodym derivatives of the functions $\int_0^T[x(t)]^md\lambda(t)(m\in\mathbb N)$ and $[\int_0^Tx(t)d\lambda(t)]^2$ on $C[0,T]$, where $\lambda$ is a complex-valued Borel measure on $[0,T]$. Finally we derive two translation theorems for the Radon-Nikodym derivatives of the functions on $C[0,T]$.

Keywords : analogue of Wiener measure, conditional Wiener integral, Feynman integral, translation theorem, Wiener integral, Wiener space