Liu process and uncertain calculus

Xiaowei Chen,Dan A Ralescu

doi:10.1186/2195-5468-1-3

Xiaowei Chen, Dan A Ralescu

Open Access

https://doi.org/10.1186/2195-5468-1-3

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Abstract

Uncertain calculus is a branch of mathematics that deals with the integral and differential of functions of uncertain processes. This paper first introduces the Liu process as an uncertain process defined by the Liu integral. Some properties of Liu processes are investigated such as sample continuity property, finite variation property, and the fact that a continuously differentiable function of the Liu process is another Liu process, among others. Based on the Liu process, the uncertain integral is extended. Furthermore, some mathematical properties are proved, including the fundamental theorem, change of variable theorem, and integration by parts theorem. Finally, uncertain calculus with respect to multiple Liu processes is discussed.

Highlights

A stochastic process, a sequence of random variables indexed by time, is a useful tool to deal with dynamical random phenomena
Based on the Wiener process, stochastic calculus was developed by Ito [2]
In 1967, the stochastic integral was extended by Kunita and Watanabe [3] to square integrable martingales

Summary

Introduction

A stochastic process, a sequence of random variables indexed by time, is a useful tool to deal with dynamical random phenomena. The theory of integration and differentiation of uncertain processes with respect to the Liu process is called the Liu calculus. The theory of integration and differentiation of uncertain processes with respect to a renewal process is called the Yao calculus.

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