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Extended Wiener Process in Nonstandard Analysis

Applied Mathematics 2020年第3期11卷 247-254页

作者： shuya Kanagawa Kiyoyuki Tchizawa Department of Mathematics Tokyo City University Setagaya Tokyo Japan Institute of Administration Engineering Ltd. Chiyoda-ku Tokyo Japan

standing on a different view point from Anderson, we prove that the extended Wiener process defined by Anderson satisfies the definition of the Wiener process in standard analysis, for example the Wiener process at time t obeys the normal distribution N(0,t) by showing the central limit theorem. The essential theory used in the proof is the extended convolution property in nonstandard analysis which is shown by Kanagawa, Nishiyama and Tchizawa (2018). When processing the extension by non-standardization, we have already pointed out that it is needed to proceed the second extension for the convolution, not only to do the first extension for the delta function. In section 2, we shall introduce again the extended convolution as preliminaries described in our previous paper. In section 3, we shall provide the extended stochastic process using a hyper number N, and it satisfies the conditions being Wiener process. In section 4, we shall give a new proof for the non-differentiability in the Wiener process.

关键词： Wiener Process Ito’s Process stochastic Differential Equation s-continuity Nonstandard Analysis

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Proof of Ito’s Formula for Ito’s Process in Nonstandard Analysis

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Applied Mathematics 2019年第7期10卷 561-567页

作者： shuya Kanagawa Kiyoyuki Tchizawa Department of Mathematics Tokyo City University Setagaya Tokyo Japan Institute of Administration Engineering Ltd. Tokyo Japan

In our previous paper [1], we proposed a non-standardization of the concept of convolution in order to construct an extended Wiener measure using nonstandard analysis by E. Nelson [2]. In this paper, we consider Ito’s integral with respect to the extended Wiener measure and extend Ito’s formula for Ito’s process. Because of doing the extension of Ito’s formula, we could treat stochastic differential equations in the sense of nonstandard analysis. In this framework, we need the nonstandardization of convolution again. It was not yet proved in the last paper, therefore we shall provide the proof.

关键词： Ito’s Process stochastic Differential Equation s-continuity Nonstandard Analysis

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Extended Wiener Measure by Nonstandard Analysis for Financial Time series

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Applied Mathematics 2018年第8期9卷 975-984页

作者： shuya Kanagawa Ryoukichi Nishiyama Kiyoyuki Tchizawa Epartment of Mathematics Tokyo City University Tokyo Japan Department of literature Ryukoku University Kyoto Japan Institute of Administration Engineering Ltd. Tokyo Japan

We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random... 详细信息

关键词： Time series Black-sholes Model s-continuity Nonstandard Analysis Delta-Function

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