# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1907.02598