A maximum principle for the stochastic differential equations with multiplicative noise

TL;DR

This paper introduces a maximum principle for stochastic differential equations with multiplicative noise, linking probability currents with Ito paths and extending an existing extremum principle.

Contribution

It presents a novel maximum principle that relates probability currents and Ito paths in stochastic differential equations with multiplicative noise, generalizing previous extremum principles.

Findings

Established a simplified forward equation for Ito paths.

Linked probability current agreement with resolving paths.

Generalized the extremum principle for stochastic systems.

Abstract

Agreement of the probability current with the resolving paths requires a simplified forward equation for the (unique) Ito paths. Their increments are the most probable rather than expected ones, in accordance with an existing extremum principle. The latter is also generalized.