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

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.