Abstract
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.
Highlights
IntroductionAnderson [1] provided stochastic processes in nonstandard analysis to show that for f : Ω → R the following condition (a) is equivalent with (b),
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)
Summary
Anderson [1] provided stochastic processes in nonstandard analysis to show that for f : Ω → R the following condition (a) is equivalent with (b),. He proved that the measure on ∗R according to the process W (t ) defined by Definition 3.2 satisfies the above condition (b). As the main theorem in this paper, we shall provide the extended stochastic process W (t ) described newly in ∗R. It satisfies the conditions being the Wiener process in nonstandard analysis. Because of the above reasons, we need a nonstandard analysis for the convolution by the hyperfunction
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