Parabolic systems and stochastic controls: nonlocality, nonlinearity, and time-inconsistency

Abstract

This thesis aims to advance the theories of partial differential equation (PDE) and stochastic differential equation (SDE), and by which, we address decade-long open problems in the field of stochastic controls. We develop systematically a theory of nonlocal parabolic systems in aspects of existence, uniqueness, stability, and computational method, where there is an external time parameter t on top of the temporal and spatial variables (s, y). The nonlocality comes from the two time variable structure. Such equations arise from time-inconsistent problems in game theory or behavioral economics, where the observations and preferences are (reference-)time-dependent. This thesis first obtains the well-posedness of nonlocal linear systems and establishes a Schauder-type prior estimate for the solutions with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. Subsequently, we take advantage of linearization methods and quasilinearization methods to establish the well-posedness results of solutions under the semilinear, quasilinear, and fully nonlinear case. Besides of pushing the frontiers of PDE, our theoretical framework allows the control variate entering the diffusion of state process, which breaks successfully through the existing bottleneck of time-inconsistent stochastic control problems. Moreover, we also provide a general and unified treatment for the Feynman-Kac formulas of a flow of forward-backward SDEs.