A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE

Title & Authors
A CHANGE OF SCALE FORMULA FOR CONDITIONAL WIENER INTEGRALS ON CLASSICAL WIENER SPACE
Yoo, Il; Chang, Kun-Soo; Cho, Dong-Hyun; Kim, Byoung-Soo; Song, Teuk-Seob;

Abstract
Let $\small{X_k(x)=({\int}^T_o{\alpha}_1(s)dx(s),...,{\int}^T_o{\alpha}_k(s)dx(s))\;and\;X_{\tau}(x)=(x(t_1),...,x(t_k))}$ on the classical Wiener space, where $\small{{{\alpha}_1,...,{\alpha}_k}}$ is an orthonormal subset of $\small{L_2}$ [0, T] and \${\tau}:0 of functions on classical Wiener space having the form $\small{G_{\gamma}(x)=F(x){\Psi}({\int}^T_ov_1(s)dx(s),...,{\int}^T_o\;v_{\gamma}(s)dx(s))}$, for $\small{F{\in}S\;and\;{\Psi}={\psi}+{\phi}({\psi}{\in}L_p(\mathbb{R}^{\gamma}),\;{\phi}{\in}\hat{M}(\mathbb{R}^{\gamma}))}$, which need not be bounded or continuous. Here S is a Banach algebra on classical Wiener space and $\small{\hat{M}(\mathbb{R}^{\gamma})}$ is the space of Fourier transforms of measures of bounded variation over $\small{\mathbb{R}^{\gamma}}$. As results of the formula, we derive a change of scale formula for the conditional Wiener integrals $\small{E[G_{\gamma}|X_{\tau}]\;and\;E[F|X_{\tau}]}$. Finally, we show that the analytic Feynman integral of F can be expressed as a limit of a change of scale transformation of the conditional Wiener integral of F using an inversion formula which changes the conditional Wiener integral of F to an ordinary Wiener integral of F, and then we obtain another type of change of scale formula for Wiener integrals of F.
Keywords
change of scale formula;conditional analytic Feynman integral;conditional analytic Wiener integral;conditional Wiener integral;
Language
English
Cited by
1.
A CHANGE OF SCALE FORMULA FOR GENERALIZED WIENER INTEGRALS II, Journal of the Chungcheng Mathematical Society, 2013, 26, 1, 111
2.
SCALE TRANSFORMATIONS FOR PRESENT POSITION-INDEPENDENT CONDITIONAL EXPECTATIONS, Journal of the Korean Mathematical Society, 2016, 53, 3, 709
3.
Analogues of conditional Wiener integrals and their change of scale transformations on a function space, Journal of Mathematical Analysis and Applications, 2009, 359, 2, 421
4.
Integral Transforms on a Function Space with Change of Scales Using Multivariate Normal Distributions, Journal of Function Spaces, 2016, 2016, 1
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