A constructive approach to existence of equilibria in time-inconsistent stochastic control problems

TL;DR

This paper develops a method to prove the existence of equilibria in a broad class of mean-field time-inconsistent stochastic control problems using discretization, BSDEs, and weak convergence, with numerical implementation via finite Markov chains.

Contribution

It extends equilibrium construction techniques to mean-field time-inconsistent problems and introduces a discretization approach with numerical methods.

Findings

Existence of equilibria proved for a large class of problems.

Method applies weak convergence of n-person games.

Numerical approximation via finite Markov chains demonstrated.

Abstract

We extend the construction of equilibria for linear-quadratic and mean-variance portfolio problems available in the literature to a large class of mean-field time-inconsistent stochastic control problems in continuous time. Our approach relies on a time discretization of the control problem via n-person games, which are characterized via the maximum principle using Backward Stochastic Differential Equations (BSDEs). The existence of equilibria is proved by applying weak convergence arguments to the solutions of n-person games. A numerical implementation is provided by approximating n-person games using finite Markov chains.