Stochastic differential equations: loss of the Markov property by multiplicative noise

TL;DR

This paper investigates how solutions to stochastic differential equations with multiplicative noise lose the Markov property, providing analytical tools to understand their behavior and density evolution over time.

Contribution

It reveals the conditions under which solutions retain Markovian properties and introduces formulas for density peak movement and shape evolution in multiplicative noise SDEs.

Findings

Solutions are non-Markovian with multiplicative noise

On coarse scales, solutions are Markovian only in the anti-Ito case

Provides criteria for steady state attainment

Abstract

The solutions of SDEs with multiplicative noise are not Markovian. On a coarse-grained time scale they still are, but only in the "anti-Ito" case. This allows a simple computation of the most likely path. Any density peak moves along such a path, and its shape evolves according to further analytical formulas. This even provides some new insights into the asymptotic densities for large times, e.g. the criterion for attaining a quiescent steady state.