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

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.

💡 Research Summary

The paper investigates a partially observed stochastic linear Stackelberg differential game in which both Brownian motions and Poisson random measures drive the dynamics, and the players’ objectives are formulated under a mean‑variance (MV) criterion. The leader (player 2) moves first, the follower (player 1) observes a noisy signal Y₁ and reacts, while the leader observes a different noisy signal Y₂. Both signals are linear functions of the hidden state X and are corrupted by independent Brownian noises and compensated Poisson jumps. The control processes u₁(t) and u₂(t) are required to be adapted to the filtrations generated by Y₁ and Y₂ respectively, which creates a circular dependency between the controls and the unobservable state.

To break this circularity, the authors adopt the orthogonal decomposition technique introduced by Sun and Xiong (2024). They decompose the hidden state X(t) into its conditional mean ˆX(t)=E