AN EVALUATION FORMULA FOR A GENERALIZED CONDITIONAL EXPECTATION WITH TRANSLATION THEOREMS 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 < ··· < t n < t n +1 = T of [0 ,T ], define X n : C [0 ,T ] → R n +1 by X n ( x ) = ( x ( t 0 ) ,x ( t 1 ) ,...,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 (cid:82) T 0 [ x ( t )] m dλ ( t )( m ∈ N ) and [ (cid:82) T 0 x ( t ) dλ ( t )] 2 on C [0 ,T ], where λ 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 ].