Relation of a New Interpretation of Stochastic Differential Equations to Ito Process

TL;DR

This paper explores a new interpretation of stochastic differential equations that incorporates a Boltzmann-Gibbs distribution, clarifies its relation to Ito processes, and highlights its potential for revealing both local and global dynamics.

Contribution

It establishes a clear, testable relation between the new SDE interpretation and classical Ito processes, enhancing understanding of stochastic dynamics in arbitrary dimensions.

Findings

Derived a concise relation between the new interpretation and Ito process.

Showed the new interpretation includes a Boltzmann-Gibbs distribution.

Highlighted the potential to reveal local and global dynamics.

Abstract

Stochastic differential equations (SDE) are widely used in modeling stochastic dynamics in literature. However, SDE alone is not enough to determine a unique process. A specified interpretation for stochastic integration is needed. Different interpretations specify different dynamics. Recently, a new interpretation of SDE is put forward by one of us. This interpretation has a built-in Boltzmann-Gibbs distribution and shows the existence of potential function for general processes, which reveals both local and global dynamics. Despite its powerful property, its relation with classical ones in arbitrary dimension remains obscure. In this paper, we will clarify such connection and derive the concise relation between the new interpretation and Ito process. We point out that the derived relation is experimentally testable.