Time-inconsistent recursive zero-sum stochastic differential games

Qingmeng Wei,Zhiyong Yu

doi:10.3934/mcrf.2018045

Abstract

In this paper, a kind of time-inconsistent recursive zero-sum stochastic differential game problems are studied by a hierarchical backward sequence of time-consistent subgames. The notion of feedback control-strategy law is adopted to constitute a closed-loop formulation. Instead of the time-inconsistent saddle points, a new concept named equilibrium saddle points is introduced and investigated, which is time-consistent and can be regarded as a local approximate saddle point in a proper sense. Moreover, a couple of equilibrium Hamilton-Jacobi-Bellman-Isaacs equations are obtained to characterize the equilibrium values and construct the equilibrium saddle points.

Highlights

Let (Ω, F, F, P) be a complete filtered probability space on which a d-dimensional standard Brownian motion W (·) is defined, and F = {Ft}t≥0 is its natural filtration
For any t ∈ [0, T ] regraded as an initial time, we denote the set of all possible initial states by
As the same as the one given in the optimal control theory, for any (t, ξ) ∈ D, an admissible feedback control law for Player 1 is a measurable mapping u1 : [t, T ] × Rn → U1 such that for each u2(·) ∈ U2[t, T ], there exists a unique solution to the following SDE:

Summary

Introduction

As the same as the one given in the optimal control theory (see YongZhou [20]), for any (t, ξ) ∈ D, an admissible feedback control law for Player 1 is a measurable mapping u1 : [t, T ] × Rn → U1 such that for each u2(·) ∈ U2[t, T ], there exists a unique solution to the following SDE: dX (r). We shall develop the multi-person differential games approach (which is for time-inconsistent optimal control problems) to a new one called backward sequence of time-consistent subgames to investigate Problem (InC-SDG).

We notice that

For any