Similarity Between Two Stochastic Differential Systems

TL;DR

This paper investigates the degree of similarity between two stochastic differential systems using homeomorphic mappings, providing theoretical conditions and applications to extend conjugacy concepts in stochastic dynamics.

Contribution

It introduces a novel approach to measure similarity between stochastic systems via minimizers of homeomorphic mappings, with new theoretical conditions and applications.

Findings

Established necessary and sufficient conditions for the existence of the minimizer K*

Provided a stochastic maximum principle related to system similarity

Applied results to stochastic Hartman-Grobman theorem

Abstract

The main focus of this paper is to explore how much similarity between two stochastic differential systems. Motivated by the conjugate theory of stochastic dynamic systems, we study the relationship between two systems by finding homeomorphic mappings $K$. Particularly, we use the minimizer $K^*$ to measure the degree of similarity. Under appropriate assumptions, we give sufficient and necessary conditions for the existence of the minimizer $K^*$. The former result can be regarded as a strong law of large numbers, while the latter is a stochastic maximum principle. Finally, we provide different examples of stochastic systems and an application to stochastic Hartman Grobman theorem. Thus, the results illustrate what is the similarity, extending the conjugacy in stochastic dynamical systems.