Linear Quadratic Stackelberg Stochastic Differential Games: Closed-Loop Solvability

TL;DR

This paper investigates the conditions for the existence of closed-loop solutions in linear-quadratic Stackelberg stochastic differential games with deterministic coefficients, using Riccati equations and backward stochastic differential equations.

Contribution

It introduces the concept of closed-loop solvability for these games and characterizes optimal strategies through Riccati and backward stochastic differential equations.

Findings

Follower's optimal strategy characterized by Riccati and BSDE.

Necessary conditions for leader's strategy via coupled Riccati equations.

Sufficiency remains open due to limitations of the completion-of-square method.

Abstract

This paper is concerned with the closed-loop solvability of one kind of linear-quadratic Stackelberg stochastic differential game, where the coefficients are deterministic. The notion of the closed-loop solvability is introduced, which require to be independent of the initial state. The follower's problem is solved first, and the closed-loop optimal strategy is characterized by a Riccati equation, together with an adapted solution to a linear backward stochastic differential equation. Then the necessary conditions of the existence of the leader's nonanticipating closed-loop optimal strategy is obtained via a system of cross-coupled Riccati equations. The sufficiency is open since the completion-of-square method is invalid.