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An overlapping information linear-quadratic Stackelberg stochastic differential game with two leaders and two followers

Yu Si, Jingtao Shi

Abstract

This paper is concerned with an overlapping information linear-quadratic (LQ) Stackelberg stochastic differential game with two leaders and two followers, where the diffusion terms of the state equation contain both the control and state variables. A distinct feature lies in that, the noisy information available to the leaders and the followers may be asymmetric and have overlapping part. Using a coupled system of Riccati equations, the followers first solve an LQ nonzero-sum stochastic differential Nash game with partial information, and then the leaders solve a partial information LQ nonzero-sum stochastic differential Nash game driven by a conditional mean-field type forward-backward stochastic differential equation (CMF-FBSDE). By maximum principle, completion of squares and decoupling methods, the state-estimate feedback representation of the Stackelberg-Nash equilibrium is obtained.

An overlapping information linear-quadratic Stackelberg stochastic differential game with two leaders and two followers

Abstract

This paper is concerned with an overlapping information linear-quadratic (LQ) Stackelberg stochastic differential game with two leaders and two followers, where the diffusion terms of the state equation contain both the control and state variables. A distinct feature lies in that, the noisy information available to the leaders and the followers may be asymmetric and have overlapping part. Using a coupled system of Riccati equations, the followers first solve an LQ nonzero-sum stochastic differential Nash game with partial information, and then the leaders solve a partial information LQ nonzero-sum stochastic differential Nash game driven by a conditional mean-field type forward-backward stochastic differential equation (CMF-FBSDE). By maximum principle, completion of squares and decoupling methods, the state-estimate feedback representation of the Stackelberg-Nash equilibrium is obtained.
Paper Structure (6 sections, 5 theorems, 112 equations)

This paper contains 6 sections, 5 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Main result
  4. Problem of the followers
  5. Problem of the leaders
  6. Conclusion

Key Result

Theorem 3.1

Under Assumption A1, $\left(u_1^*(\cdot), u_2^*(\cdot)\right)$ is a Nash equilibrium point of the followers' problem, if and only if where $\left(p_j(\cdot), k_{j1}(\cdot), k_{j2}(\cdot), k_{j3}(\cdot) \right)$, $j=1,2$ are the unique $\mathcal{F}_t$-adapted solutions satisfying BSDEs:

Theorems & Definitions (10)

  • Theorem 3.1
  • Lemma 3.1
  • Theorem 3.2
  • Remark 3.1
  • Theorem 3.3
  • Theorem 3.4
  • proof
  • proof
  • proof
  • proof