Change of Scale Formulas for a Generalized Conditional Wiener Integral

Abstract

Abstract. Let C[0, t] denote the space of real-valued continuous functions on [0, t] and define a random vector Zn :with h = 0 a.e. Using a simple formula for a conditional expectation on C[0, t] with Zn, we evaluate a generalized analytic conditional Wiener integral of the functionvr(s)dx(s)) for F in a Banach algebra and for Ψ = f + φ which need not be bounded or continuous, whereand φ is the Fourier transform of a measure of bounded variation over R r . Finally we establish various change of scale transformations for the generalized analytic conditional Wiener integrals of Gr with the conditioning function Zn.

A Conditional Fourier-Feynman Transform and Conditional Convolution Product with Change of Scales on a Function Space II

“…We now have the following relationships among the conditional Fourier-Feynman transform, the conditional convolution products, and the generalized Wiener integrals of the functions in S . Their proofs are similar to the proofs of the results in [12] with additional calculations.…”

“…where and Ψ are given by (12) and (13), respectively. Using the same method as used in the proof of Theorem 3.2 in [7] ( , ⃗ ) = ∫…”

A Conditional Fourier-Feynman Transform and Conditional Convolution Product With Change of Scales on a Function Space I

“…so that for λ > 0, y ∈ C[0, T ] and ξ n ∈ R n+1 we have by Lemma 2.3 y)), where Ψ r is given by (7). We note that if 1 ≤ p < ∞, then by the change of variable theorem…”

“…The author [4] introduced a generalized conditional Wiener integral with drift on C[0, T ] and then, derived two simple formulas which calculate the conditional expectations in terms of ordinary expectations, that is, non-conditional expectations. Using the simple formulas on C[0, T ], the author and his coauthors [5,6,7] established a conditional analytic Fourier-Feynman transform, a conditional convolution product which has no drift, and change of scale formulas for conditional Wiener integrals which simplify the evaluations of the analytic conditional Feynman integrals, because the measure used on C[0, T ] is not scale-invariant [1,2].…”

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

“…On the space C[0, t] the author in [7] derived a simple formula for a generalized conditional Wiener integral given the vector-valued conditioning function X n+1 . Using the formula with X n+1 , Yoo and the author in [12] evaluated a generalized analytic conditional Wiener integral of the function G r having the form…”