Stochastic differential equations with covariant probabilities

TL;DR

This paper discusses the covariance properties of stochastic differential equations with multiplicative noise, introduces an 'anti-Ito' interpretation, and provides explicit transformations to simplify such equations while preserving key features.

Contribution

It introduces an 'anti-Ito' framework for covariance in stochastic differential equations and explicitly describes how to remove multiplicative noise through variable changes.

Findings

Simplified Fokker-Planck equation preserving key features

Explicit change of variables to remove multiplicative noise

Covariance requires the 'anti-Ito' interpretation

Abstract

Covariance of the resulting probabilities requires the "anti-Ito" sense. The corresponding Fokker-Planck equation is simplified and preserves important features of the case with a constant diffusion. Multiplicative noise can always be removed by a change of the variables, which is specified explicitly.