Kramers–Moyal expansion

In stochastic processes, Kramers-Moyal expansion refers to a Taylor series expansion of the Master equation, named after Hans Kramers and José Enrique Moyal. This expansion transforms the integro differential Master equation to a partial differential equation. The expansion for the probability density is given by

${\frac {\partial p(x,t)}{\partial t}}=\sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n!}}{\frac {\partial ^{n}}{\partial x^{n}}}[\alpha _{n}(x)p(x,t)]$

where

$\alpha _{n}(x)=\int _{-\infty }^{\infty }(x'-x)^{n}W(x'|x)\ dx'$.

Here $W(x'|x)$ is the transition probability rate. Fokker-Planck equation is obtained by keeping only the first two terms of the series.