An Evaluation Formula for Radon–Nikodym Derivatives Similar to Conditional Expectations over Paths

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 $0=t_0<t_1<\cdots <t_n=T$ of [0, T], define $X:C[0,T]\rightarrow \mathbb R^{n+1}$ by $X(x)=(x(t_0),x(t_1),\ldots ,$ $x(t_n))$. 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 which has a drift and an initial weight. As applications of the formula, we evaluate the Radon–Nikodym derivatives of the functions $\int _0^T[x(t)]^m\mathrm{d}\lambda (t)(m\in \mathbb N)$ and $[\int _0^Tx(t)\mathrm{d}\lambda (t)]^2$ on C[0, T], where $\lambda$ is a complex-valued Borel measure on [0, T].