An Introduction to the Martingale Problem

The martingale problem is an important tool in probability theory introduced by Stroock and Varadhan (see [2], [3] and [4]) as an alternative approach to Ito’s theory of diffusion processes on R^d (see also [1] 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.

Program
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.

Rereferences:
[1] Karatzas, I.; Shreve, S. Brownian motion and stochastic calculus. Springer Science & Business Media (1991).
[2] D. W. Stroock, S. R. S. Varadhan. Diffusion processes with continuous coefficients, I. Comm. Pure Appl. Math., vol. XXII, 345–400 (1967).
[3] D. W. Stroock, S. R. S. Varadhan. Diffusion processes with continuous coefficients, II. Comm. Pure Appl. Math., vol. XXII, 479–530 (1969).
[4] D. W. Stroock, S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften vol. 233, Springer (1979).