On the Well-posedness of Hamilton-Jacobi-Bellman Equations of the Equilibrium Type

TL;DR

This paper establishes the well-posedness of a class of nonlocal Hamilton-Jacobi-Bellman equations related to time-inconsistent stochastic control, using advanced PDE techniques and providing probabilistic representations and financial applications.

Contribution

It introduces new methods to prove global and local well-posedness of nonlocal PDEs of equilibrium type with time and space nonlocality, including a probabilistic solution representation.

Findings

Proved global well-posedness using Schauder estimates and the method of continuity.

Established local well-posedness via linearization and fixed point arguments.

Provided a probabilistic representation of solutions and an example of a solvable financial time-inconsistency problem.

Abstract

This paper studies the well-posedness of a class of nonlocal parabolic partial differential equations (PDEs), or equivalently equilibrium Hamilton-Jacobi-Bellman equations, which has a strong tie with the characterization of the equilibrium strategies and the associated value functions for time-inconsistent stochastic control problems. Specifically, we consider nonlocality in both time and space, which allows for modelling of the stochastic control problems with initial-time-and-state dependent objective functionals. We leverage the method of continuity to show the global well-posedness within our proposed Banach space with our established Schauder prior estimate for the linearized nonlocal PDE. Then, we adopt a linearization method and Banach's fixed point arguments to show the local well-posedness of the nonlocal fully nonlinear case, while the global well-posedness is attainable provided that a very sharp a-priori estimate is available. On top of the well-posedness results, we also provide a probabilistic representation of the solutions to the nonlocal fully nonlinear PDEs and an estimate on the difference between the value functions of sophisticated and na\"{i}ve controllers. Finally, we give a financial example of time inconsistency that is proven to be globally solvable.