Lecturer, Markov processes
Research supervisor of Student research group "Fundamental problems in financial mathematics"
Markov processes are a universal model for a wide range of stochastic processes observed in reality. Scientific literature on Markov processes and their various generalizations is constantly growing, uncovering new applications in various fields of natural and socio-economic sciences.
From a probabilistic point of view, Markov processes are described by distributions on the space of the trajectories (which can be specified by stochastic differential equations). From an analytical point of view, these processes are defined by Markov semigroups, that determine the evolution of averages and arise as solutions to a class of differential and pseudo-differential equations with partial derivatives possessing the property of positivity preservation. Thus, the entire theory is based on a deep interaction and mutual enrichment of stochastic and functional analysis.
This course offers a compact yet systematic exposition of the fundamental ideas, methods, and applications of the theory of Markov processes. It begins with a detailed enough study of the most fundamental classes, such as Brownian motion and Levy processes, and includes the necessary preparatory material on martingale theory.
Potential students of the course are assumed to have a strong grasp of the probability theory fundamentals, but no prior knowledge in stochastic processes theory is required.See full course outline.