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Mathematics Colloquium

Xin Zhang
University of Vienna

Title: PDE in Control -- Applications in Finance and Learning
Date: Friday, January 26, 2024
Place and Time: Love 101, 3:05-3:55 pm

Abstract. The theory of stochastic control offers a framework for understanding, analyzing, and designing random systems with the goal of achieving desired outcomes. It finds wide-ranging applications in finance, engineering, and data science. Stochastic control problems are known to be related to nonlinear parabolic partial differential equations (PDEs), which are powerful tools in problem solving. In this talk, we will review the viscosity theory of nonlinear parabolic PDEs on Rd and discuss their applications in adversarial prediction problems and optimal win martingales. Subsequently, we will introduce the mean field control problem, which models the decision-making in large populations of interacting agents. This corresponds to a class of nonlinear parabolic PDEs on Wasserstein space. As a main result, we will present a comparison principle for such equations and characterize the value function of a filtering problem as the unique viscosity solution. In the end, I will highlight some potential applications of PDEs on Wasserstein space in finance and learning.