Equivalence of Stochastic Equations and Martingale Problems

The fact that the solution of a martingale problem for a diffusion process gives a weak solution of the corresponding Itô equation is well-known since the original work of Stroock and Varadhan. The result is typically proved by constructing the driving Brownian motion from the solution of the martingale problem and perhaps an auxiliary Brownian motion. This constructive approach is much more challenging for more general Markov processes where one would be required to construct a Poisson random measure from the sample paths of the solution of the martingale problem. A “soft” approach to this equivalence is presented here, which begins with a joint martingale problem for the solution of the desired stochastic equation and the driving processes and applies a Markov mapping theorem to show that any solution of the original martingale problem corresponds to a solution of the joint martingale problem. These results coupled with earlier results on the equivalence of forward equations and martingale problems show that the three standard approaches to specifying Markov processes (stochastic equations, martingale problems, and forward equations) are, under very general conditions, equivalent in the sense that existence and/or uniqueness of one implies existence and/or uniqueness for the other two.