An Introduction to the Martingale Problem
The martingale problem is an important tool in probability theory introduced by Stroock and Varadhan (see ,  and ) as an alternative approach to Ito’s theory of diffusion processes on Rd (see also  Section 5.4). The purpose of this mini course is to present an introduction to the martingale problem in general setting and its connection to Markov processes.
A solution of a martingale problem is a random path whose mechanism of evolution is determined by a so-called Markov generator. When this Markov generator is a vector field, the martingale problem reduces to the respective ode. In this sense, the martingale problem is a generalization of an ode and the Markov property can be viewed as the random version of the flow property in the ode case. Both arise as a consequence of uniqueness of solutions.
Lecture 1: The martingale formulation.
Lecture 2: Regular conditional probability distributions and martingales.
Lecture 3: A Markov-like property for the martingale problem.
Lecture 4: Uniqueness: some examples.
 Karatzas, I.; Shreve, S. Brownian motion and stochastic calculus. Springer Science & Business Media (1991).
 D. W. Stroock, S. R. S. Varadhan. Diffusion processes with continuous coefficients, I. Comm. Pure Appl. Math., vol. XXII, 345–400 (1967).
 D. W. Stroock, S. R. S. Varadhan. Diffusion processes with continuous coefficients, II. Comm. Pure Appl. Math., vol. XXII, 479–530 (1969).
 D. W. Stroock, S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften vol. 233, Springer (1979).