Stochastic differential equations: loss of the Markov property by multiplicative noise
arXiv:1606.07464
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.
The arguments in the Sections 3.2 and 3.3 are not conclusive, and the Markov property is not disproved. Many other statements are though correct, see arXiv:2011.11476v4 ("Revisiting the stochastic differential equations with multiplicative noise")