Past lectures

Abstract

The theory of martingales is a powerful tool for studying properties of stopping times which are of great importance in risk theory, mathematical finance, statistical sequential analysis etc. The course will provide the detailed survey of classical results and some recent developments in the theory of martingales and its applications. We shall start from the fundamental properties of martingales continued by illustrations through different stochastic models (with discrete and continuous time parameter) motivated by applications. Most parts of the course will be covered in detailed outlines, assuming only an introductory knowledge of the theory of stochastic processes.

The course will cover the following topics:

1. Basic properties of martingales. Tail asymptotics of quadratic characteristics with applications to some gambling problems.

2. Exponential martingales and measure transformations.

3. Survey of results on boundary crossing probabilities for stochastic processes. Applications to pricing of exotic options.

4. Bounds and asymptotic approximations for ruin probabilities in stochastic investment models.

5. Elements of theory of optimal stopping problems. Applications to pricing of American options.

6. Sequential approach to statistical problems. Sequential estimation and change-point problems.