Linear Quadratic Leader-follower Stochastic Differential Games: Closed-Loop Solvability

TL;DR

This paper investigates a leader-follower stochastic differential game with linear dynamics and quadratic costs, establishing conditions for closed-loop strategies via Riccati equations and backward stochastic differential equations.

Contribution

It introduces a framework for solving leader-follower stochastic differential games using Riccati equations and backward stochastic differential equations, with explicit conditions for closed-loop strategy existence.

Findings

Follower's optimal strategy characterized by Riccati and BSDE solutions

Leader's optimal control involves solving a forward-backward SDE and Riccati equation

Existence conditions for closed-loop strategies are established

Abstract

In this paper, a leader-follower stochastic differential game is studied for a linear stochastic differential equation with a quadratic cost functional. The coefficients in the state equation and the weighting matrices in the cost functionals are all deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. The follower first solves a stochastic linear quadratic optimal control problem, and his optimal closed-loop strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the leader turns to solve a stochastic linear quadratic optimal control problem of a forward-backward stochastic differential equation, necessary conditions for the existence of optimal closed-loop strategies for the leader is given by the existence of a Riccati equation. Some examples are also given.