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Linear-quadratic mixed Stackelberg-zero-sum game for mean-field regime switching system

Pengyan Huang, Na Li, Zuo Quan Xu, Harry Zheng

Abstract

Motivated by a product pricing problem, a linear-quadratic Stackelberg differential game for a regime switching system involving one leader and two followers is studied. The two followers engage in a zero-sum differential game, and both the state system and the cost functional incorporate a conditional mean-field term. Applying continuation method and induction method, we first establish the existence and uniqueness of a conditional mean-field forward-backward stochastic differential equation with Markovian switching. Based on it, we prove the unique solvability of Hamiltonian systems for the two followers and the leader. Moreover, utilizing stochastic maximum principle, decoupling approach and optimal filtering technique, the optimal feedback strategies of two followers and leader are obtained. Employing the theoretical results, we solve a product pricing problem with some numerical simulations.

Linear-quadratic mixed Stackelberg-zero-sum game for mean-field regime switching system

Abstract

Motivated by a product pricing problem, a linear-quadratic Stackelberg differential game for a regime switching system involving one leader and two followers is studied. The two followers engage in a zero-sum differential game, and both the state system and the cost functional incorporate a conditional mean-field term. Applying continuation method and induction method, we first establish the existence and uniqueness of a conditional mean-field forward-backward stochastic differential equation with Markovian switching. Based on it, we prove the unique solvability of Hamiltonian systems for the two followers and the leader. Moreover, utilizing stochastic maximum principle, decoupling approach and optimal filtering technique, the optimal feedback strategies of two followers and leader are obtained. Employing the theoretical results, we solve a product pricing problem with some numerical simulations.

Paper Structure

This paper contains 11 sections, 10 theorems, 153 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Problem formulation
  4. Literature review and contributions of this paper
  5. Notation
  6. Unique solvability of CMF-FBSDE with regime switching
  7. LQ Stackelberg game for stochastic system with regime switching
  8. Optimal strategies for two followers
  9. Optimal strategy for the leader
  10. Product pricing problem
  11. Conclusion and outlook

Key Result

Theorem 2.1

Let Assumptions (A1)-(A2) hold with $(\Psi,\Phi,\Gamma)$. Then, CMF-FBSDE fbsde admits a unique solution $\pi=\left(x_1,y_1,z_1,\tilde{z}_1,x_2,y_2,z_2,\tilde{z}_2\right)\in \left(N_{\mathbb F}^2 \left(0,T; \mathbb R^{3n}\right)\times \mathcal{M}_{\mathbb F}^2\left(0,T; \mathbb R\right)\right)^2$. M with where $K$ is a constant depending on $T$, the Lipschitz constants, $\mu_1,\nu_1,\mu_2$, and t

Figures (5)

  • Figure 1: Numerical solution of $\alpha_t$.
  • Figure 2: Numerical solutions of $p(t)$ and $y(t)$.
  • Figure 3: Numerical solutions of $u^*_{F,1}(t)$ and $u^*_{F,2}(t)$.
  • Figure 4: Numerical solution of $u^*_L(t)$.
  • Figure 5: Numerical solution of $x^*(t)$.

Theorems & Definitions (13)

  • Theorem 2.1
  • Definition 1
  • Theorem 3.1
  • Theorem 3.2
  • Remark 3.1
  • Lemma 3.1
  • Theorem 3.3
  • Theorem 3.4
  • Remark 3.2
  • Proposition 5.1
  • ...and 3 more