Time-inconsistent stochastic optimal control problems and backward stochastic volterra integral equations

¹ Department of Mathematics, National University of Singapore, Singapore 119076, Singapore. This author is supported by Singapore MOE AcRF Grants R-146-000-271-112.
² Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA. This author is supported in part by NSF Grant DMS-1812921.

^* Corresponding author: hxwang14@fudan.edu.cn

Received: 22 March 2020
Accepted: 4 March 2021

Abstract

An optimal control problem is considered for a stochastic differential equation with the cost functional determined by a backward stochastic Volterra integral equation (BSVIE, for short). This kind of cost functional can cover the general discounting (including exponential and non-exponential) situations with a recursive feature. It is known that such a problem is time-inconsistent in general. Therefore, instead of finding a global optimal control, we look for a time-consistent locally near optimal equilibrium strategy. With the idea of multi-person differential games, a family of approximate equilibrium strategies is constructed associated with partitions of the time intervals. By sending the mesh size of the time interval partition to zero, an equilibrium Hamilton–Jacobi–Bellman (HJB, for short) equation is derived, through which the equilibrium value function and an equilibrium strategy are obtained. Under certain conditions, a verification theorem is proved and the well-posedness of the equilibrium HJB is established. As a sort of Feynman–Kac formula for the equilibrium HJB equation, a new class of BSVIEs (containing the diagonal value Z(r, r) of Z(⋅ , ⋅)) is naturally introduced and the well-posedness of such kind of equations is briefly presented.