• 충청수학회지
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• 제26권1호
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• pp.111-123
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• 2013

Cameron and Storvick discovered change of scale formulas for Wiener integrals on classical Wiener space. Yoo and Skoug extended this result to an abstract Wiener space. In this paper, we investigate a change of scale formula for generalized Wiener integrals of various functions using the generalized Fourier-Feynman transform.

• 대한수학회지
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• 제53권3호
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• pp.709-723
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• 2016

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.

• 충청수학회지
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• 제24권3호
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• pp.517-528
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• 2011

Cameron and Storvick introduced change of scale formulas for Wiener integrals of bounded functions in the Banach algebra $\mathcal{S}$ of analytic Feynman integrable functions on classical Wiener space. Yoo and Skoug extended this result to an abstract Wiener space. Also Yoo, Song, Kim and Chang established a change of scale formula for Wiener integrals of functions on abstract Wiener space which need not be bounded or continuous. In this paper, we investigate a change of scale formula for generalized Wiener integrals of various functions on classical Wiener space.

• Korean Journal of Mathematics
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• 제24권3호
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• pp.489-494
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• 2016

The purpose of this paper is to treat the generalized conditional Yeh-Wiener integral for the sample path-valued conditioning function. As a special case of our results, we obtain the results in [6].

• 대한수학회보
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• 제36권4호
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• pp.809-822
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• 1999

In this paper, we introduce conditional Yeh-Wiener in-tegrals for generalized conditioning functions including vector-valued functions. And also we establish various evaluation formulas of conditional Yeh-Wiener integrals for generalized conditioning functions.

• 대한수학회지
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• 제40권1호
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• pp.73-86
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• 2003

In this paper, we define a generalized sequential operator-valued function space integral by using a generalized Wiener measure. It is an extention of the sequential operator-valued function space integral introduced by Cameron and Storvick. We prove the existence of this integral for functionals which involve some product Borel measures.

• 충청수학회지
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• 제26권4호
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• pp.735-742
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• 2013

In this note, we prove the translation theorem for the generalized analogue of Wiener measure space and we show some properties of the generalized analogue of Wiener measure from it.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제4권1호
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• pp.47-60
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• 1997

In the theory of the conditional Wiener integral, the integrand is a functional of the standard Wiener process. In this paper we consider a conditional function space integral for functionals of more general stochastic process and the generalized Kac-Feynman integral equation. We first show that the existence of a partial differential equation. We then show that the generalized Kac-Feynman integral equation is equivalent to the partial differential equation.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제14권4호
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• pp.335-354
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• 2007

In this paper, we define an analogue of generalized Wiener measure and investigate its basic properties. We define (${\hat}It{o}$ type) stochastic integrals with respect to the generalized Wiener process and prove the ${\hat}It{o}$ formula. The existence and uniqueness of the solution of stochastic differential equation associated with the generalized Wiener process is proved. Finally, we generalize the linear filtering theory of Kalman-Bucy to the case of a generalized Wiener process.

• 대한수학회보
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• 제28권2호
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• pp.147-161
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• 1991

In this paper we define the conditional generalized Wiener measure and then express the conditional generalized Wiener integral over this new measure. In particular we consider a conditional expectation of functionals of the generalized Brownian paths under the condition that the paths pass through the given points .xi.$_{1}$, .xi.$_{2}$, .., .xi.$_{n}$ at times t$_{1}$, t$_{2}$, .., t$_{n}$, respectively.ely.