The Martingale Problem

The martingale problem due to Stroock and Varadhan provides another way to define a solution of a stochastic differential equation. It is a concept that is unique to stochastic differential equation in the sense that it has no counterpart in the theory of ordinary and partial differential equations. Under this approach, existence and uniqueness of solutions of stochastic differential equations can be proved under milder conditions. First, we obtain equivalent formulations of martingale problems, and then proceed to establish existence of a solution to the martingale problem. Uniqueness of solutions is shown using certain analytical tools and Laplace transforms. Further extensions and the Markov property of solutions are discussed.