Nonlocal Fully Nonlinear Parabolic Differential Equations Arising in Time-Inconsistent Problems

Abstract

We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point $(t,s,y)$ but also at the diagonal line of the time domain $(s,s,y)$. Such equations arise from time-inconsistent problems in game theory or behavioural economics, where the observations and preferences are (reference-)time-dependent. To address the open problem of the well-posedness of the corresponding nonlocal PDEs (or the time-inconsistent problems), we first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the well-posedness of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we analogously establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.