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arxiv: 2604.04478 · v1 · submitted 2026-04-06 · 🧮 math.OC

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

Yongpeng Lin , Qingxin Meng , Maoning Tang This is my paper

Pith reviewed 2026-05-10 20:06 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic gamesNash equilibriumlinear quadraticstochastic Riccati equationsforward-backward stochastic difference equationsindefinite weighting matricesrandom coefficientsopen-loop strategies
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The pith

Necessary and sufficient conditions for open-loop Nash equilibria in discrete-time LQ stochastic games with random coefficients are derived via convex variational analysis and a global nonnegativity condition on indefinite weights.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines two-person nonzero-sum linear quadratic stochastic games in discrete time with random coefficients. It derives necessary and sufficient conditions for the existence of open-loop Nash equilibria using convex variational analysis. When the weighting matrices are indefinite, a global nonnegativity condition is introduced to restore sufficiency of the first-order conditions. The equilibria are characterized explicitly through fully coupled forward-backward stochastic difference equations and a system of non-symmetric stochastic Riccati equations with constraints. Sufficient conditions of positive semidefiniteness of the Riccati operators and structural non-degeneracy guarantee the invertibility of a related operator, allowing a closed-loop feedback representation of the equilibria.

Core claim

Using convex variational analysis, the authors derive necessary and sufficient conditions for the existence of open-loop Nash equilibria in two-person nonzero-sum linear quadratic stochastic games with random coefficients. When weighting matrices are indefinite, a global nonnegativity condition restores sufficiency. The equilibria are characterized by fully coupled forward-backward stochastic difference equations and non-symmetric stochastic Riccati equations, with positive semidefiniteness and structural non-degeneracy ensuring the invertibility for closed-loop feedback representations.

What carries the argument

Fully coupled forward-backward stochastic difference equations (FBSΔEs) and the associated system of non-symmetric stochastic Riccati equations with constraints that decouple the stochastic Hamiltonian system, with invertibility guaranteed by positive semidefiniteness of the Riccati operators and structural non-degeneracy.

If this is right

  • The first-order conditions become sufficient for optimality under the global nonnegativity condition even when the weighting matrices are indefinite.
  • The open-loop Nash equilibria admit a closed-loop feedback representation when the Riccati matrix operators are positive semidefinite and the system is structurally non-degenerate.
  • The presence of fully random coefficients leads to fully nonlinear higher-order backward stochastic difference equations in the Riccati framework, unlike algebraic Riccati equations in deterministic cases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The variational analysis and Riccati framework could be adapted to derive equilibria in continuous-time versions of these games or in games with more than two players.
  • Numerical solution methods for the resulting nonlinear backward stochastic difference equations would enable practical computation of the feedback strategies in applications.
  • The structural non-degeneracy condition might be verifiable or relaxed in models where the noise structure has specific properties, such as independent increments.

Load-bearing premise

The global nonnegativity condition on the indefinite weighting matrices together with the positive semidefiniteness and structural non-degeneracy assumptions required to guarantee invertibility of the operator and well-posedness of the FBSΔEs.

What would settle it

A low-dimensional explicit example where the weighting matrices are indefinite, violate the global nonnegativity condition, satisfy first-order conditions, but no open-loop Nash equilibrium exists, or where the Riccati operator fails to be invertible.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript studies discrete-time two-person nonzero-sum linear-quadratic stochastic games with random coefficients. Using convex variational analysis, it derives necessary and sufficient conditions for open-loop Nash equilibria. For indefinite weighting matrices, a global nonnegativity condition is introduced to restore sufficiency. Equilibria are characterized explicitly via fully coupled forward-backward stochastic difference equations (FBSΔEs) and a system of non-symmetric stochastic Riccati equations. A key contribution is the provision of sufficient conditions—positive semidefiniteness of the Riccati matrix operators together with structural non-degeneracy—that guarantee invertibility of a related operator and thereby ensure well-posedness of the closed-loop feedback representation of the open-loop Nash strategies. Fully random coefficients produce fully nonlinear higher-order backward stochastic difference equations, in contrast to algebraic Riccati equations in the deterministic case.

Significance. If the stated conditions and invertibility results hold, the work extends stochastic game theory by accommodating indefinite costs and fully random coefficients in discrete time, yielding explicit sufficient conditions for well-posedness of the associated FBSΔE/Riccati system. The treatment of nonlinear higher-order backward stochastic difference equations arising from random coefficients is a distinctive technical feature.

major comments (2)
  1. [Title] Title: The title announces viscosity solutions of Hamilton–Jacobi–Bellman equations for control systems driven by Teugels martingales, yet the abstract and manuscript develop discrete-time LQ games, FBSΔEs, and Riccati equations. This mismatch is load-bearing for the central claim, as the invertibility conditions and well-posedness results cannot attach to the titled problem without an explicit bridge between the two programs.
  2. [Abstract] Abstract (central claim paragraph): The sufficient conditions (positive semidefiniteness of Riccati operators plus structural non-degeneracy) are asserted to guarantee invertibility and closed-loop well-posedness, but the abstract supplies no explicit operator definition, no verification that the global nonnegativity condition is independent of these assumptions, and no proof outline. The load-bearing nature of this claim for the FBSΔE framework requires the full manuscript to supply the missing derivation steps.
minor comments (2)
  1. [Abstract] The sentence 'equations (FBSΔEs) with constraints. that decouple' contains a period and capitalization error; rephrase for grammatical correctness.
  2. [Introduction] Notation such as FBSΔEs and the precise meaning of 'structural non-degeneracy' should be introduced with a forward reference to the relevant section in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. The comments highlight important issues of clarity and consistency that we address below. We plan to revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Title] Title: The title announces viscosity solutions of Hamilton–Jacobi–Bellman equations for control systems driven by Teugels martingales, yet the abstract and manuscript develop discrete-time LQ games, FBSΔEs, and Riccati equations. This mismatch is load-bearing for the central claim, as the invertibility conditions and well-posedness results cannot attach to the titled problem without an explicit bridge between the two programs.

    Authors: We agree that the current title does not match the manuscript content. The title appears to have been carried over from a separate project on viscosity solutions for Teugels martingales. The present work concerns discrete-time LQ stochastic games with random coefficients, FBSΔEs, and constrained Riccati equations. We will change the title to accurately reflect the scope: “Necessary and Sufficient Conditions for Open-Loop Nash Equilibria in Discrete-Time Linear-Quadratic Stochastic Games with Random Coefficients”. This revision removes any ambiguity and ensures the claims attach directly to the developed framework. revision: yes

  2. Referee: [Abstract] Abstract (central claim paragraph): The sufficient conditions (positive semidefiniteness of Riccati operators plus structural non-degeneracy) are asserted to guarantee invertibility and closed-loop well-posedness, but the abstract supplies no explicit operator definition, no verification that the global nonnegativity condition is independent of these assumptions, and no proof outline. The load-bearing nature of this claim for the FBSΔE framework requires the full manuscript to supply the missing derivation steps.

    Authors: The abstract is a concise summary and therefore omits explicit operator definitions and full proof outlines; these are developed rigorously in the body of the manuscript (Sections 3–5), where the operator whose invertibility is established is defined, the global nonnegativity condition is shown to be independent of the semidefiniteness and non-degeneracy assumptions, and the derivation steps for well-posedness of the closed-loop representation are provided. To improve readability of the central claim, we will add one sentence to the abstract briefly naming the operator and noting that independence and invertibility are proved in the main text. The existing manuscript already contains the required derivations. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on independent variational conditions and stated assumptions

full rationale

The paper derives necessary and sufficient conditions for open-loop Nash equilibria via convex variational analysis on the discrete-time LQ game. The global nonnegativity condition on indefinite weighting matrices is introduced explicitly as an additional assumption to restore sufficiency when classical first-order conditions fail; it is not defined in terms of the equilibria or equilibria existence. The positive semidefiniteness of Riccati operators and structural non-degeneracy are presented as independent sufficient conditions guaranteeing operator invertibility and FBSΔE well-posedness, rather than being fitted to or redefined by the target closed-loop representation. No self-citation chains, ansatz smuggling, or renaming of known results appear in the provided derivation steps. The title-abstract mismatch raises scope questions but does not create a circular reduction within the mathematical argument itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard assumptions of linear dynamics and quadratic costs plus new technical conditions introduced to handle indefinite weights and random coefficients.

axioms (2)
  • domain assumption Convexity of the cost functionals for variational analysis
    Invoked to obtain necessary and sufficient conditions for open-loop Nash equilibria.
  • ad hoc to paper Well-posedness of the FBSΔEs under the stated invertibility conditions
    The existence of solutions to the coupled system is asserted once the operator invertibility is guaranteed by positive semidefiniteness and non-degeneracy.

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