Fokker-Planck and Chapman-Kolmogorov equations for Ito processes with finite memory

TL;DR

This paper extends the classical Fokker-Planck and Chapman-Kolmogorov equations to include Ito processes with finite memory, demonstrating that these equations still hold and providing a Gaussian example with one memory state.

Contribution

It shows that finite memory can be incorporated into the Fokker-Planck and Chapman-Kolmogorov framework, expanding their applicability beyond Markov processes.

Findings

Finite memory processes satisfy K1, K2, and Chapman-Kolmogorov equations.

A Gaussian transition density with one memory state is constructed.

A new proof that the Black-Scholes Green function describes a Martingale.

Abstract

The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker-Planck pde), and show that a Chapman-Kolmogorov eqn. follows as well. We adapt Friedman's derivation to emphasize that finite memory is not excluded. We then give an example of a Gaussian transition density with 1 state memory satisfying both K1, K2, and the Chapman-Kolmogorov eqns. We begin the paper by explaining the meaning of backward diffusion, and end by using our interpretation to produce a new, short proof that the Green function for the Black-Scholes pde describes a Martingale in the risk neutral discounted stock price.