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Viscosity Solutions of Hamilton--Jacobi--Bellman Equations for Control Systems Driven by Teugels Martingales

Yongpeng Lin, Qingxin Meng, Maoning Tang

Abstract

This paper studies discrete-time two-person nonzero-sum linear quadratic stochastic games with random coefficients. Using convex variational analysis, we derive necessary and sufficient conditions for the existence of open-loop Nash equilibria. When weighting matrices are indefinite, the classical first-order conditions are no longer sufficient for optimality; we introduce a global nonnegativity condition to restore sufficiency, which becomes a cornerstone of the subsequent analysis. To characterize the equilibria explicitly, we develop fully coupled forward-backward stochastic difference equations and a system of non-symmetric stochastic Riccati equations (FBS$Δ$Es) with constraints. that decouple the stochastic Hamiltonian system. A key technical contribution is the provision of sufficient conditions -- positive semidefiniteness of the Riccati matrices operators and structural non-degeneracy -- that guarantee the invertibility of a related operator, ensuring the well-posedness of the closed-loop feedback representation of the open-loop Nash equilibrium strategies. A distinctive feature of this work is the presence of fully random coefficients, which leads to fully nonlinear higher-order backward stochastic difference equations in the Riccati framework, in contrast to the algebraic Riccati equations in the deterministic setting.

Viscosity Solutions of Hamilton--Jacobi--Bellman Equations for Control Systems Driven by Teugels Martingales

Abstract

This paper studies discrete-time two-person nonzero-sum linear quadratic stochastic games with random coefficients. Using convex variational analysis, we derive necessary and sufficient conditions for the existence of open-loop Nash equilibria. When weighting matrices are indefinite, the classical first-order conditions are no longer sufficient for optimality; we introduce a global nonnegativity condition to restore sufficiency, which becomes a cornerstone of the subsequent analysis. To characterize the equilibria explicitly, we develop fully coupled forward-backward stochastic difference equations and a system of non-symmetric stochastic Riccati equations (FBSEs) with constraints. that decouple the stochastic Hamiltonian system. A key technical contribution is the provision of sufficient conditions -- positive semidefiniteness of the Riccati matrices operators and structural non-degeneracy -- that guarantee the invertibility of a related operator, ensuring the well-posedness of the closed-loop feedback representation of the open-loop Nash equilibrium strategies. A distinctive feature of this work is the presence of fully random coefficients, which leads to fully nonlinear higher-order backward stochastic difference equations in the Riccati framework, in contrast to the algebraic Riccati equations in the deterministic setting.

Paper Structure

This paper contains 11 sections, 18 theorems, 306 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Construction of Teugels Martingales and the Martingale Representation Theorem
  4. Backward stochastic differential equations (BSDEs) with jumps
  5. $L^2$ a priori estimates for SDEs and BSDEs
  6. DPP for Stochastic Optimal Control of FBSDEs with Lévy Jumps
  7. Problem formulation and well-posedness of solutions
  8. The Dynamic Programming Principle
  9. The Existence of Viscosity Solutions
  10. The Uniqueness of Viscosity Solutions
  11. Conclusion

Key Result

Lemma 2.1

Under the above framework, any square-integrable $\mathcal{F}_T$-measurable random variable $F$ admits the unique representation $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (36)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: Comparison
  • Remark 2.1
  • Lemma 2.4
  • Remark 2.2
  • Lemma 2.5
  • Remark 2.3
  • Proposition 3.1
  • proof
  • ...and 26 more