Linear-Quadratic Stochastic Stackelberg Differential Games for Jump-Diffusion Systems

TL;DR

This paper develops a comprehensive framework for solving linear-quadratic stochastic Stackelberg differential games involving jump-diffusion systems with random coefficients, providing explicit solutions for leader and follower strategies.

Contribution

It introduces a novel approach to solve the indefinite LQ stochastic control problem for the leader in jump-diffusion systems, extending the Four-Step Scheme to this complex setting.

Findings

Explicit state-feedback solutions for follower and leader controls.

Characterization of Stackelberg equilibrium in jump-diffusion systems.

Handling of the technical challenges posed by jump processes.

Abstract

This paper considers linear-quadratic (LQ) stochastic leader-follower Stackelberg differential games for jump-diffusion systems with random coefficients. We first solve the LQ problem of the follower using the stochastic maximum principle and obtain the state-feedback representation of the open-loop optimal solution in terms of the integro-stochastic Riccati differential equation (ISRDE), where the state-feedback type control is shown to be optimal via the completion of squares method. Next, we establish the stochastic maximum principle for the indefinite LQ stochastic optimal control problem of the leader using the variational method. However, to obtain the state-feedback representation of the open-loop solution for the leader, there is a technical challenge due to the jump process. To overcome this limitation, we consider two different cases, in which the state-feedback type optimal control for the leader in terms of the ISRDE can be characterized by generalizing the Four-Step Scheme. Finally, in these two cases, we show that the state-feedback representation of the open-loop optimal solutions for the leader and the follower constitutes the Stackelberg equilibrium. Note that the (indefinite) LQ control problem of the leader is new and nontrivial due to the coupled FBSDE constraint induced by the rational behavior of the follower.