A simple formula for an analogue of conditional Wiener integrals and its applications

Abstract

Abstract. Let C [0, T ] denote the space of real-valued continuous functions on the interval [0, T ] and for a partition τ :In this paper, with the conditioning function X τ , we derive a simple formula for conditional expectations of functions defined on C[0, T ] which is a probability space and a generalization of Wiener space. As applications of the formula, we evaluate the conditional expectation of functions of the formfor x ∈ C[0, T ] and derive a translation theorem for the conditional expectation of integrable functions defined on the space C[0, T ].

Integration With Respect to Analogue of Wiener Measure Over Paths in Wiener Space and Its Applications

“…Cho established the simple formula for conditional expectation in C 0 (B) case [1]. Our result is very similar to previous results but the process of its proof is not quite same.…”

Operator-Valued Function Space Integrals via Conditional Integrals on an Analogue Wiener Space Ii

“…Let X n be given by (4). Moreover let ϕ r be normally distributed with the mean vector 0 ∈ R r and the nontrivial variance-covariance matrix α 2 I r , where α > 0 and I r is the r-dimensional identity matrix.…”

“…Corollary 3.6. Let n ≥ 2, X n be given by (4) Using the same method as used in the proof of Theorem 3.5, we can prove the following theorem.…”

“…Let n ≥ 2, the assumptions be as given in Lemma 2.2 and X n be given by (4) [5]. We note that such a function which is not a normal density can be obtained.…”

“…Further work extending the above L(L 1 , L ∞ )-theory with the conditional analytic Feynman integrals was studied by the author [5] over the space (C r [0, t], w r ϕ ) [9,11] of the continuous R r -valued paths on [0, t] which generalizes the space C r 0 [0, t]. In fact the author [4] introduced the conditional Wiener integral over C r [0, t] and derived a simple formula for the conditional Wiener integral with the conditioning function X n : C r [0, t] → R (n+1)r given by X n (x) = (x(t 0 ), x(t 1 ), . .…”