Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

Let denote a generalized Wiener space, the space of real-valued continuous functions on the interval and define a stochastic process by for and , where with a.e. and is continuous on . Let random vectors and be given by and , where is a partition of . In this paper we derive a translation theorem for a generalized Wiener integral and then prove that is a generalized Brownian motion process with drift . Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions on with the drift and the conditioning functions and . As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions on . 1. Introduction Let denote the Wiener space, the space of real-valued continuous functions on with . On the space, Yeh [1] introduced an inversion formula that a conditional expectation can be found by a Fourier-transform. But Yeh’s inversion formula is very complicated in its application when the conditioning function is vector-valued. In [2], Park and Skoug derived a simple formula for conditional Wiener integrals on with a vector-valued conditioning function given by , where is a partition of the interval . In their simple formula, they expressed the conditional Wiener integral directly in terms of an ordinary Wiener integral. Using the simple formula in [2], Chang and Skoug [3] investigated the effect that drift has on the conditional Fourier-Feynman transform, the conditional convolution product, and various relationships that occur between them. On the other hand, let denote the space of real-valued continuous functions on the interval . Im and Ryu [4] introduced a probability measure on , where is a probability measure on the Borel class of . When , the Dirac measure concentrated at , is exactly the Wiener measure on . On the space , the author [5, 6] derived two simple formulas for the conditional Wiener -integral of functions on with the vector-valued conditioning functions and given by and which generalize the Park and Skoug’s formula in [2]. Using these formulas with the conditioning functions and , he evaluated the conditional Wiener -integral of function of the form for any positive integer . Let with a.e. on , and let be a continuous function on . Define a stochastic process by for and . Let and be given by In this paper, we derive a translation theorem for a generalized Wiener -integral, and then prove that is a generalized Brownian motion process with drift and variance for . Furthermore, we derive two simple formulas for generalized conditional Wiener integrals