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

Jingtao Lin, Jingtao Shi

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.

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

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.
Paper Structure (8 sections, 15 theorems, 94 equations)

This paper contains 8 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. The follower's problem
  4. The leader's problem
  5. Filtering equations
  6. The leader's optimal control problem
  7. Concluding remarks
  8. Appendix: Non-linear filtering

Key Result

Lemma 3.1

Let Assumption assumption state hold. For a fixed $u_2\in\mathcal{U}^L_{ad}$, and for any $u_1\in\mathcal{U}^F_{ad}$, let $X^{u_1,u_2}$ be the corresponding state process satisfying state equation 2.1. Then $\hat{X}^{u_1,u_2}$ evolves according to and $\tilde{X}^{u_1,u_2}$ satisfies Further, $\Sigma(t)\coloneqq\mathbb{E}\left[(\tilde{X}^{u_1,u_2}(t))(\tilde{X}^{u_1,u_2}(t))^\top\vert\mathcal{Y}^

Theorems & Definitions (27)

  • Remark 2.1
  • Lemma 3.1
  • Remark 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • Remark 4.1
  • ...and 17 more