A PDE approach for open-loop equilibriums in time-inconsistent stochastic optimal control problems

TL;DR

This paper develops a PDE-based framework for identifying open-loop equilibrium strategies in complex time-inconsistent stochastic control problems involving jump-diffusions, extending existing methods with new integro-PDE systems.

Contribution

It introduces a novel PDE approach for open-loop equilibria in jump-diffusion models, including a verification theorem and application to portfolio optimization.

Findings

Derived two systems of integro-PDEs for equilibrium strategies

Proved a verification theorem providing sufficient conditions

Applied the framework to a mean-variance portfolio problem

Abstract

This paper studies open-loop equilibriums for a general class of time-inconsistent stochastic control problems under jump-diffusion SDEs with deterministic coefficients. Inspired by the idea of Four-Step-Scheme for forward-backward stochastic differential equations with jumps (FBSDEJs, for short), we derive two systems of integro-partial differential equations (IPDEs, for short). Then, we rigorously prove a verification theorem which provides a sufficient condition for open-loop equilibrium strategies. As an illustration of the general theory, we discuss a mean-variance portfolio selection problem under a jump-diffusion model.