A generalized simple formula for evaluating Radon-Nikodym derivatives over paths
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Let $C[0,T]$ denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. Define $Z_{\vec e,n}:C[0,T]\to\mathbb R^{n+1}$ by \begin{align*} Z_{\vec e,n}(x)=\left(x(0),\int_0^Te_1(t)dx(t),\ldots,\int_0^Te_n(t)dx(t)\right), \end{align*} where $e_1, \ldots,e_n$ are of bounded variations on $[0,T]$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $Z_{\vec e,n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on $C[0,T]$ which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval $[0,T]$.
Abstract
Let $C[0,T]$ denote a generalized analogue of Wiener space, the space of real-valued continuous functions on the interval $[0,T]$. Define $Z_{\vec e,n}:C[0,T]\to\mathbb R^{n+1}$ by \begin{align*} Z_{\vec e,n}(x)=\left(x(0),\int_0^Te_1(t)dx(t),\ldots,\int_0^Te_n(t)dx(t)\right), \end{align*} where $e_1, \ldots,e_n$ are of bounded variations on $[0,T]$. In this paper we derive a simple evaluation formula for Radon-Nikodym derivatives similar to the conditional expectations of functions on $C[0,T]$ with the conditioning function $Z_{\vec e,n}$ which has an initial weight and a kind of drift. As applications of the formula, we evaluate the Radon-Nikodym derivatives of various functions on $C[0,T]$ which are of interested in Feynman integration theory and quantum mechanics. This work generalizes and simplifies the existing results, that is, the simple formulas with the conditioning functions related to the partitions of time interval $[0,T]$.
참고문헌 (12)
- R. H. Cameron, "Analytic functions" (1980), Springer
- C. C. Chen, "Principles and Techniques in Combinatorics" (1992), World Scientific Publishing Co., Inc
- Dong Cho, "Measurable functions similar to the itô integral and the Paley-Wiener-Zygmund integral over continuous paths", Filomat, 32(18), pp.6441~6456 (2018), National Library of Serbia[DOI]
- Dong Hyun Cho, "An Evaluation Formula for Radon–Nikodym Derivatives Similar to Conditional Expectations over Paths", Bulletin of the Malaysian Mathematical Sciences Society, 44(1), pp.203~222 (2021), Springer Science and Business Media LLC[DOI]
- 조동현, "A Banach algebra and its equivalent spaces over paths with a positive measure", 대한수학회논문집, 35(3), pp.809~823 (2020), 대한수학회[DOI]
- 조동현, "An evaluation formula for a generalized conditional expectation with translation theorems over paths", 대한수학회지, 57(2), pp.451~470 (2020), 대한수학회[DOI]
- C. Park, "A simple formula for conditional Wiener integrals with applications", Pacific J. Math, 135(2), pp.381~394 (1988)
- C. Park, "Conditional Wiener integrals. II", Pacific J. Math, 167(2), pp.293~312 (1995)
- I. D. Pierce, "On a family of generalized Wiener spaces and applications" (2011), ProQuest LLC
- 류근식, "THE GENERALIZED ANALOGUE OF WIENER MEASURE SPACE AND ITS PROPERTIES", 호남수학학술지, 32(4), pp.633~642 (2010), 호남수학회
- 류근식, "The translation theorem on the generalized analogue of Wiener space and its applications", 충청수학회지, 26(4), pp.735~742 (2013), 충청수학회[DOI]
- J. Yeh, "Inversion of conditional expectations", Pacific J. Math, 52, pp.631~640 (1974)