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.