Analogues of Conditional Wiener Integrals with Drift and Initial Distribution on a Function Space

Abstract EN

Let C [ 0 , T ] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [ 0 , T ], and define a stochastic process Z : C [ 0 , T ] × [ 0 , T ] → R by Z ( x , t ) = ∫ 0 t h ( u ) d x ( u ) + x ( 0 ) + a ( t ) , for x ∈ C [ 0 , T ] and t ∈ [ 0 , T ] , where h ∈ L 2 [ 0 , T ] with h ≠ 0 a.e.

and a is a continuous function on [ 0 , T ] .

Let Z n : C [ 0 , T ] → R n + 1 and Z n + 1 : C [ 0 , T ] → R n + 2 be given by Z n ( x ) = ( Z ( x , t 0 ) , Z ( x , t 1 ) , … , Z ( x , t n ) ) and Z n + 1 ( x ) = ( Z ( x , t 0 ) , Z ( x , t 1 ) , … , Z ( x , t n ) , Z ( x , t n + 1 ) ) , where 0 = t 0 < t 1 < ⋯ < t n < t n + 1 = T is a partition of [ 0 , T ] .

In this paper we derive two simple formulas for generalized conditional Wiener integrals of functions on C [ 0 , T ] with the conditioning functions Z n and Z n + 1 which contain drift and initial distribution.

As applications of these simple formulas we evaluate generalized conditional Wiener integrals of the function e x p { ∫ 0 T Z ( x , t ) d m L ( t ) } including the time integral on C [ 0 , T ] .