SCALE TRANSFORMATIONS FOR PRESENT POSITION-INDEPENDENT CONDITIONAL EXPECTATIONS

Title & Authors
SCALE TRANSFORMATIONS FOR PRESENT POSITION-INDEPENDENT CONDITIONAL EXPECTATIONS
Cho, Dong Hyun;

Abstract
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 $\small{Z_n:C[0,t}$$\small{]}$$\small{{\rightarrow}{\mathbb{R}}^n}$ by $\small{Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))}$, where 0 < $\small{t_1}$ < $\small{{\cdots}}$ < $\small{t_n}$ < t is a partition of [0, t] and $\small{h{\in}L_2[0,t}$$\small{]}$$\small{}$ with $\small{h{\neq}0}$ 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 $\small{Z_n}$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $\small{F(x)=f(\int_{0}^{t}e(s)dx(s))}$ for $\small{x{\in}C[0,t}$$\small{]}$$\small{}$, where $\small{f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})}$ and e is a unit element in $\small{L_2[0,t}$$\small{]}$$\small{}$. 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 $\small{L_2[0,t}$$\small{]}$$\small{}$ used in the transformation is independent of e and the conditioning function $\small{Z_n}$ does not contain the present positions of the generalized Wiener paths.
Keywords
analytic conditional Feynman integral;analytic conditional Wiener integral;conditional Wiener integral;Wiener integral;Wiener space;
Language
English
Cited by
References
1.
R. H. Cameron, The translation pathology of Wiener space, Duke Math. J. 21 (1954), 623-627.

2.
R. H. Cameron and W. T. Martin, The behavior of measure and measurability under change of scale in Wiener space, Bull. Amer. Math. Soc. 53 (1947), 130-137.

3.
R. H. Cameron and D. A. Storvick, Change of scale formulas for Wiener integral, Rend. Circ. Mat. Palermo (2) Suppl. 17 (1987), 105-115.

4.
R. H. Cameron and D. A. Storvick, Some Banach algebras of analytic Feynman integrable functionals, Lecture Notes in Math. 798, Springer, Berlin-New York, 1980.

5.
D. H. Cho, Change of scale formulas for conditional Wiener integrals as integral transforms over Wiener paths in abstract Wiener space, Commun. Korean Math. Soc. 22 (2007), no. 1, 91-109.

6.
D. H. Cho, A simple formula for a generalized conditional Wiener integral and its applications, Int. J. Math. Anal. 7 (2013), no. 29, 1419-1431.

7.
D. H. Cho, Scale transformations for present position-dependent conditional expectations over continuous paths, (2015), preprint.

8.
D. H. Cho, B. J. Kim, and I. Yoo, Analogues of conditional Wiener integrals and their change of scale transformations on a function space, J. Math. Anal. Appl. 359 (2009), no. 2, 421-438.

9.
M. K. Im and K. S. Ryu, An analogue of Wiener measure and its applications, J. Korean Math. Soc. 39 (2002), no. 5, 801-819.

10.
B. S. Kim, Relationship between the Wiener integral and the analytic Feynman integral of cylinder function, J. Chungcheong Math. Soc. 27 (2014), no. 2, 249-260.

11.
I. Yoo, K. S. Chang, D. H. Cho, B. S. Kim, and T. S. Song, A change of scale formula for conditional Wiener integrals on classical Wiener space, J. Korean Math. Soc. 44 (2007), no. 4, 1025-1050.

12.
I. Yoo and D. H. Cho, Change of scale formulas for a generalized conditional Wiener integral on a function space, Bull. Iranian Math. Soc. (2015), submitted.

13.
I. Yoo and D. L. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces, Internat. J. Math. Math. Sci. 17 (1994), no. 2, 239-247.

14.
I. Yoo and D. L. Skoug, A change of scale formula for Wiener integrals on abstract Wiener spaces II, J. Korean Math. Soc. 31 (1994), no. 1, 115-129.

15.
I. Yoo, T. S. Song, B. S. Kim, and K. S. Chang, A change of scale formula for Wiener integrals of unbounded functions, Rocky Mountain J. Math. 34 (2004), no. 1, 371-389.