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

Cho, Dong Hyun

doi:10.1007/s10587-009-0030-6

Cited by 12 publications

(13 citation statements)

References 12 publications

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“…In Refs. [6] and [7], Cho derived two simple formulas for the conditional w ϕ -integrals of the functions on C [0, t] with the vector-valued conditioning functions X n and X n+1 which are defined on C [0, t]. These formulas express the conditional w ϕ -integrals directly in terms of non-conditional w ϕ -integrals.…”

Section: Introductionmentioning

confidence: 99%

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Analogues of conditional Wiener integrals and their change of scale transformations on a function space

Cho

Kim

Yoo

2009

Journal of Mathematical Analysis and Applications

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Keywords:Change of scale transformation Conditional analytic Feynman w ϕ -integral Conditional analytic Wiener w ϕ -integral Conditional w ϕ -integral Simple formula for conditional w ϕ -integralIn this paper, using simple formulas for conditional w ϕ -integrals on C [0, t] with the conditioning functions X n and X n+1 , we evaluate the conditional analytic Wiener w ϕ -integrals of the function G r having the formfor F ∈ S wϕ which is a Banach algebra, and for Ψ = f + φ which need not be bounded or continuous, where f ∈ L p (R r ) and φ ∈M(R r ). And also, we establish several change of scale transformations for the conditional analytic Wiener w ϕ -integrals of G r with the conditioning functions X n and X n+1 .

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Section: Introductionmentioning

confidence: 99%

“…In the following two theorems, we introduce simple formulas for conditional w ϕ -integrals on C [0, t] [6,7].…”

Section: Introductionmentioning

confidence: 99%

Analogues of conditional Wiener integrals and their change of scale transformations on a function space

Cho

Kim

Yoo

2009

Journal of Mathematical Analysis and Applications

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“…Im and Ryu [4] introduced a probability measure on [0, ], where is a probability measure on the Borel class of R. When = 0 , the Dirac measure concentrated at 0, is exactly the Wiener measure on 0 [0, ]. On the space [0, ], the author [5,6] derived two simple formulas for the conditional Wienerintegral of functions on [0, ] with the vector-valued conditioning functions : [0, ] → R +1 and +1 : [0, ] → R +2 given by ( ) = ( ( 0 ), ( 1 ), . .…”

Section: Introductionmentioning

confidence: 99%

Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

Cho

2013

Journal of Function Spaces and Applications

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LetC[0,t]denote a generalized Wiener space, the space of real-valued continuous functions on the interval[0,t]and define a stochastic processY:C[0,t]×[0,t]→ℝbyY(x,s)=∫0s‍h(u)dx(u)+a(s)forx∈C[0,t]ands∈[0,t], whereh∈L2[0,t]withh≠0a.e. andais continuous on[0,t]. Let random vectorsYn:C[0,t]→ℝnandYn+1:C[0,t]→ℝn+1be given byYn(x)=(Y(x,t1),…,Y(x,tn))andYn+1(x)=(Y(x,t1),…,Y(x,tn),Y(x,tn+1)), where0<t1<⋯<tn<tn+1=tis a partition of[0,t]. In this paper we derive a translation theorem for a generalized Wiener integral and then prove thatYis a generalized Brownian motion process with drifta. Furthermore, we derive two simple formulas for generalized conditional Wiener integrals of functions onC[0,t]with the drift and the conditioning functionsYnandYn+1. As applications of these simple formulas, we evaluate the generalized conditional Wiener integrals of various functions onC[0,t].

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“…In the following theorem, we introduce a simple formula for the conditional w ϕ -integrals on C[0, t] [7].…”

Section: Introductionmentioning

confidence: 99%

A Time-Independent Conditional Fourier-Feynman Transform and Convolution Product on an Analogue of Wiener Space

Cho¹

2013

Honam Mathematical Journal

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Abstract. Let C[0, t] denote the function space of all real-valued continuous paths on [0, t]. Define Xn :, where 0 = t0 < t1 < · · · < tn < t is a partition of [0, t]. In the present paper, using a simple formula for the conditional expectation given the conditioning function Xn, we evaluate the Lp(1 ≤ p ≤ ∞)-analytic conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions which have the formwhere {v1, · · · , vr} is an orthonormal subset of L2[0, t] and f ∈ Lp(R r ). We then investigate several relationships between the conditional Fourier-Feynman transform and the conditional convolution product of the cylinder functions.

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

Cited by 12 publications

References 12 publications

Analogues of conditional Wiener integrals and their change of scale transformations on a function space

Analogues of conditional Wiener integrals and their change of scale transformations on a function space

Evaluation Formulas for Generalized Conditional Wiener Integrals with Drift on a Function Space

A Time-Independent Conditional Fourier-Feynman Transform and Convolution Product on an Analogue of Wiener Space

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