Ito Processes with Finitely Many States of Memory

We show that Ito processes imply the Fokker-Planck (K2) and Kolmogorov backward time (K1) partial differential eqns. (pde) for transition densities, which in turn imply the Chapman-Kolmogorov equation without approximations. This result is not restricted to Markov processes. We define ‘finite memory’ and show that Ito processes admit finitely many states of memory. We then provide an example of a Gaussian transition density depending on two past states that satisfies both K1, K2, and the Chapman-Kolmogorov eqn. Finally, we show that transition densities of Black-Scholes type pdes with finite memory are martingales and also satisfy the Chapman-Kolmogorov equation. This leads to the shortest possible proof that the transition density of the Black-Scholes pde provides the so-called ‘martingale measure’ of option pricing.