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

TL;DR

This paper develops a comprehensive framework for analyzing stochastic differential games with random coefficients, introducing new conditions and equations to characterize Nash equilibria in complex stochastic control systems.

Contribution

It introduces a global nonnegativity condition for indefinite weighting matrices and develops coupled stochastic Riccati equations for explicit equilibrium characterization.

Findings

Necessary and sufficient conditions for Nash equilibria derived

Global nonnegativity condition restores sufficiency in indefinite cases

Fully coupled stochastic Riccati equations developed for explicit solutions

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$\Delta$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.