Dynamical Approaches for Some Complex Stochastic Systems

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The principal investigators will undertake several research projects with a special focus on dynamic optimization/game problems involving complex stochastic systems, as well as effective mathematical tools for solving these problems. The research on the general mean field master equations introduces new methodologies that will lead to a powerful tool for studying large interacting particle systems appearing in numerous applications in economics, finance (especially systemic risk), social science, and engineering. The research concerning the set-valued stochastic dynamical systems develops some new technical tools and has intrinsic connection to front propagation and mean curvature motions. Some new concepts in the project are expected to fundamentally change the current framework of set-valued stochastic analysis. The part concerning the Kyle-Back strategic insider equilibrium model brings new perspectives to an important problem in the market macrostructure theory as well as the concept of dynamic Markov bridges. All the projects that will be pursued have direct connections to applied fields such as stochastic control/game and stochastic finance/economics. The PIs will continue actively involving Ph.D students and postdoc fellows in research, disseminating research findings through conferences, and strengthening the connections with local financial communities through a colloquium series.The first part of the research introduces new methodologies to find the crucial monotonicity conditions for general mean field games, which will ensure the uniqueness of the mean field equilibrium and lead to the global well-posedness of the associated master equation, an infinite dimensional PDE or PDE system that characterizes the dynamics of the value function. The second part of the research focuses on the characterization of stochastic dynamic systems whose values are “sets”, motivated by several signature cases including the problems in the first part when uniqueness of equilibria fails, the time-inconsistent problems studied in previous research, and the issue of dynamic multivariate (systemic) risk measures. The new theory of set-valued PDEs and set-valued Backward SDEs, along with several new notions such as Itô’s formula for set-valued functions and a new set-valued stochastic integral desirable for the set-valued martingale representation will be considered, and are expected to have fundamental impact to the existing set-valued stochastic analysis. The third part of the research concerns the Kyle-Back equilibrium model with dynamic information. A new stochastic two-point boundary value problem will be considered, as a theoretical basis for finding the equilibrium under a fairly general model of underlying assets that allows interaction among the different agents in the market.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.