Kristoffer Lindensjö: time-inconsistent stochastic control - a (classical) PDE characterization

Abstract
Time-inconsistent stochastic control is a game-theoretic generalization of standard stochastic control, with well-known applications in economics and finance.

One of the most important results of standard (time-consistent) stochastic control is the characterization of the optimal control and the (optimal) value function as the solution to a (deterministic) PDE known as the Hamilton-Jacobi-Bellman equation (HJB).

Naturally, one would hope that time-inconsistent stochastic control problems offer a similar possibility. Indeed, Björk, Khapko & Murgoci (2016) introduce a system of PDEs, the extended HJB system, and prove a verification theorem saying that if the extended HJB system has a (classical) solution then this solution is in fact the equilibrium value function and the equilibrium control.

The main result of the present paper is that we (under the assumptions of the existence of an equilibrium control and ad hoc regularity) show that the equilibrium value function and the equilibrium control are a (classical) solution to the extended HJB system.

The setting is that of a general (Markovian) Itô diffusion, and a general time-inconsistent bequest function.

Keywords: Time-inconsistent stochastic control, Dynamic inconsistency, Dynamic programming, Dynamic programming principle, Hamilton-Jacobi-Bellman equation, extended HJB system, Differential games, Nash equilibrium, Time inconsistent preferences