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

Abstract

The usual derivation of the Fokker-Planck partial differential eqn. (pde) assumes the Chapman-Kolmogorov equation for a Markov process [1,2]. Starting instead with an Ito stochastic differential equation (sde), 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 [3] 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 time diffusion, and end by using our interpretation to produce a very short proof that the Green function for the Black-Scholes pde describes a Martingale in the risk neutral discounted stock price.Stochastic process; martingale; Ito process; stochastic differential eqn.; memory; nonMarkov process; 2 backward time diffusion; Fokker-Planck; Kolmogorov’s partial differential eqns.; Chapman-Kolmogorov eqn.; Black- Scholes eqn