Mathematical Finance Seminar with Sergey Nadtochiy: Probabilistic Solutions to Stefan Equations

Speaker: Sergey Nadtochiy, associate professor of applied mathematics, Illinois Institute of Technology

Title: Probabilistic Solutions to Stefan Equations

Abstract: This talk is concerned with the probabilistic methods for solving Stefan free-boundary PDEs (a.k.a. laplacian growth models). The latter equations appear in many models of fundamental physical and biological processes, such as: phase transition (i.e., melting/freezing), phase segregation (e.g., aging of alloys), crystal growth, neurons interaction, etc.. Despite their importance, to date, there exists no general existence and uniqueness theory for such equations due to the potential singularity of the solutions. Recently, the probabilistic methods, based on the analysis of associated mean-field particle systems and McKean-Vlasov equations, were successfully used to tackle the mathematical challenges that could not be addressed by the classical analytic methods, yielding new well-posedness results for certain types of Stefan equations. I will present an overview of the existing theory, including the recent existence and uniqueness results for solutions to Stefan equation with surface tension. This talk is based on joint works with F. Delarue, M. Shkolnikov, X. Zhang, and Y. Guo.

 

Mathematic Finance Seminar