Invariants of Fokker-Planck equations

TL;DR

This paper introduces a framework for identifying invariants of Fokker-Planck equations, enabling the reduction of fluctuation growth in stochastic systems, with applications demonstrated in financial share price modeling.

Contribution

It provides a general formula for the time evolution of invariants' fluctuations and applies this to finance to control fluctuation growth.

Findings

Invariant expectation remains constant over time.

The theory reduces the growth rate of fluctuations.

Application demonstrated in share price modeling.

Abstract

A weak invariant of a stochastic system is defined in such a way that its expectation value with respect to the distribution function as a solution of the associated Fokker-Planck equation is constant in time. A general formula is given for time evolution of fluctuations of the invariants. An application to the problem of share price in finance is illustrated. It is shown how this theory makes it possible to reduce the growth rate of the fluctuations.