Kanagawa, S., Nishiyama, R. and Tchizawa, K. (2018) Extended Wiener Measure by Nonstandard Analysis for Financial Time Series. Applied Mathematics, 9, 975-984.
https://doi.org/10.4236/am.2018.98066

has been cited by the following article:

• TITLE: Proof of Ito’s Formula for Ito’s Process in Nonstandard Analysis

  AUTHORS: Shuya Kanagawa, Kiyoyuki Tchizawa

  KEYWORDS: Ito’s Process, Stochastic Differential Equation, S-Continuity, Nonstandard Analysis

  JOURNAL NAME: Applied Mathematics, Vol.10 No.7, July 22, 2019

  ABSTRACT: 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.