Scale transformations for present position-independent conditional expectations
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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 $Z_n : C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\ldots,\int_0^{t_n}h(s) dx(s))$, 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 $f\in L_p (\mathbb R)(1\le p\le\infty)$ 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.
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 $Z_n : C[0,t]\to\mathbb R^n$ by $Z_n(x)=(\int_0^{t_1}h(s) dx(s),\ldots,\int_0^{t_n}h(s) dx(s))$, 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 $f\in L_p (\mathbb R)(1\le p\le\infty)$ 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.
참고문헌 (15)
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