Mini-course by professor of Saint Petersburg State University Mikhail Anatolyevich Lifshits “Some Problems on Pursuit and Approximation of Random Processes”

As part of the Summer School of Stochastic Analysis-2025, Professor of St. Petersburg State University, Doctor of Physical and Mathematical Sciences Mikhail Anatolyevich Lifshits gave a mini-course “Some Problems on Pursuit and Approximation of Random Processes”

We consider the problem of approximation of a non-smooth random process by a smoother one.
Formally, we minimize a sort of energy (an integral functional of the derivative of the approximating process) under some constraints on the distance (uniform or average one) between the approximated and the approximating processes.
If the whole trajectory of the approximated process is available (the so-called non-adaptive setting), we have a rather usual approximation problem. If only the past  trajectory of the approximated process is available (the adaptive setting), then one has a pursuit problem that essentially belongs to the optimal control theory.
The lectures survey the results obtained during the last decade by the lecturer in collaboration with I.Ibragimov, Z.Kabluchko, E.Setterqvist and some lecturer's students.

Mini-course Lifshits