A Partially Observed Stochastic Linear Stackelberg Differential Game with Poisson Jumps under Mean-Variance Criteria

TL;DR

This paper develops a novel approach to solve partially observed stochastic linear Stackelberg differential games with Poisson jumps under mean-variance criteria, using orthogonal decomposition and stochastic filtering techniques.

Contribution

It introduces a new method combining orthogonal decomposition and non-linear stochastic filtering to address circular dependencies in partially observed Stackelberg games with jumps.

Findings

Derived observable state feedback Stackelberg equilibria.

Formulated Riccati equations for equilibrium strategies.

Extended stochastic filtering to Poisson jump processes.

Abstract

In this paper, a partially observed stochastic linear Stackelberg differential game with mean-variance criteria is studied. Randomness comes from Brownian motions and Poisson random measures. which leads to a circular dependency. We follow the orthogonal decomposition method to overcome the circular dependency of the control and state processes. Both original problems of the follower and leader are decomposed into several fully observed problems with mean-variance criteria. During these processes, non-linear stochastic filtering with Poisson random measures, developed in this paper, plays an important role. Besides the follower's problem is embedded into a class of auxiliary stochastic linear-quadratic optimal control problem of stochastic differential equations with Poisson jumps, the leader's problem is also embedded into a class of auxiliary stochastic linear-quadratic optimal control problem of forward-backward stochastic differential equations with Poisson jumps. Observable state feedback Stackelberg equilibria are obtained, via some Riccati equations.