Partial and full solutions of stochastic differential equations

TL;DR

This paper discusses the properties of the anti-Ito integral in stochastic differential equations, highlighting its correctness in mean shift and deriving a simplified Fokker-Planck equation applicable to Brownian motion with arbitrary friction.

Contribution

It introduces the 'full' solutions for stochastic differential equations, emphasizing the unique role of the anti-Ito integral and deriving a simplified Fokker-Planck equation for complex Brownian motion.

Findings

Anti-Ito integral correctly shifts the mean in stochastic equations.

The 'full' Fokker-Planck equation applies to Brownian motion with arbitrary friction.

Backward equation coincides with the Fokker-Planck in noise contribution.

Abstract

Only the "anti-Ito" integral yields the correct shift of the mean, by the fact that the elements of its Riemannian sum hold in the order O(dt) rather than only in O(sqrt dt). The corresponding "full" Fokker-Planck equation is particularly simple and the only one applying for Brownian motion with an arbitrary friction law. The "full" backward equation coincides with it in the noise contribution.