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arxiv: 2603.07428 · v4 · submitted 2026-03-08 · 🧮 math.OC · math.PR

Constrained zero-sum LQ differential games for jump-diffusion systems with random coefficients

Yanyan Tang , Xun Li , Jie Xiong This is my paper

Pith reviewed 2026-05-15 15:37 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords zero-sum differential gamesstochastic LQ controljump-diffusionRiccati equationsconstrained controlsregime switchingFBSDEsopen-loop solvability
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The pith

Closed-loop representations for constrained zero-sum LQ games with jumps come from indefinite extended Riccati equations.

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

The paper establishes that under a uniform convexity-concavity condition, cone-constrained zero-sum linear-quadratic differential games driven by jump-diffusion processes with random coefficients admit open-loop saddle points. These saddle points are first characterized using forward-backward stochastic differential equations, but due to the constraints the classical explicit solution methods do not work. By applying Meyer's Ito formula and completing the square, the authors obtain an explicit closed-loop form for the saddle point in terms of solutions to a new set of multidimensional indefinite extended stochastic Riccati equations with jumps. This matters because it provides a way to compute feedback strategies for competitive stochastic control problems that include sudden regime shifts and constraints, which appear in applications like portfolio management or robust control.

Core claim

Under the uniform convexity-concavity condition, the open-loop solvability of the cone-constrained two-player zero-sum stochastic linear-quadratic differential game for jump-diffusion systems is established and the saddle point is characterized via FBSDEs. However, to obtain an explicit expression, Meyer's Ito formula combined with completing the square is used to derive a closed-loop representation based on solutions to new multidimensional indefinite extended stochastic Riccati equations with jumps. For a special case, the existence of solutions to these equations is proved.

What carries the argument

The multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs), which allow the derivation of the closed-loop saddle point representation for the constrained game by solving the associated matrix-valued equations that account for jumps and random coefficients.

If this is right

  • Open-loop saddle points exist for the game when the UCC condition holds.
  • The saddle point has a closed-loop representation in terms of IESREJ solutions.
  • Solutions to the IESREJs exist in special cases of the system.
  • The FBSDE characterization holds but requires the new Riccati approach for explicitness due to constraints.

Where Pith is reading between the lines

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

  • The completing-the-square technique may generalize to other stochastic games with constraints and discontinuous noise.
  • Numerical methods for solving the IESREJs could enable practical implementation in high-dimensional problems.
  • This framework might connect to mean-field game approximations when the number of players increases.
  • Regime switching combined with jumps suggests applications in modeling abrupt economic shifts in competitive settings.

Load-bearing premise

The uniform convexity-concavity condition must hold for the game to admit an open-loop saddle point, without which the FBSDE characterization and Riccati reduction may not hold.

What would settle it

Finding a specific instance of the game where the UCC condition is met but the corresponding IESREJs have no solution, or where the proposed closed-loop controls fail to form a saddle point in direct simulation.

read the original abstract

This paper investigates a cone-constrained two-player zero-sum stochastic linear-quadratic (SLQ) differential game for stochastic differential equations (SDEs) with regime switching and random coefficients driven by a jump-diffusion process. Under the uniform convexity-concavity (UCC) condition, we establish the open-loop solvability of the game and characterize the open-loop saddle point via the forward-backward stochastic differential equations (FBSDEs). However, since both controls are constrained, the classical four-step scheme fails to provide an explicit expression for the saddle point. To overcome this, by employing Meyer's It\^o formula together with the method of completing the square, we derive a closed-loop representation for the open-loop saddle point based on solutions to a new kind of multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs). Furthermore, for a special case, we prove the existence of solutions to IESREJs.

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 paper investigates a cone-constrained two-player zero-sum stochastic linear-quadratic differential game for jump-diffusion SDEs with regime switching and random coefficients. Under the uniform convexity-concavity (UCC) condition, it establishes open-loop solvability and characterizes the saddle point via FBSDEs. It then derives a closed-loop representation of the open-loop saddle point by applying Meyer's Itô formula and completing the square, expressed in terms of solutions to a new class of multidimensional indefinite extended stochastic Riccati equations with jumps (IESREJs). Existence of solutions to the IESREJs is proven only for a special case.

Significance. If the existence of solutions to the general IESREJs can be established, the work would provide a useful explicit representation for constrained indefinite stochastic LQ games with jumps and random coefficients, extending classical Riccati methods via standard stochastic calculus tools. The FBSDE characterization under UCC and the reduction to IESREJs are technically sound where they apply, and the special-case existence result is a positive step toward constructivity.

major comments (2)
  1. [Main derivation and existence result (as summarized in the abstract)] The central closed-loop representation (derived via Meyer's Itô formula and completing the square) is expressed in terms of solutions to the IESREJs, but the manuscript proves existence of these equations only for a special case. This leaves the general result for random coefficients conditional on an unverified assumption, which is load-bearing for the claim of an explicit saddle-point representation.
  2. [FBSDE characterization and IESREJ reduction] The FBSDE characterization of the open-loop saddle point under the UCC condition is standard, but the subsequent reduction to IESREJs for the constrained case requires the existence result to be constructive; without general existence, the four-step scheme failure noted in the paper is not fully overcome for the stated problem class.
minor comments (2)
  1. [Notation and equation definitions] Clarify the precise definition and multidimensional structure of the IESREJs at the first point of introduction, including how the indefinite nature and jump terms enter the equation.
  2. [Existence section] The special-case existence proof would benefit from an explicit statement of the assumptions that are relaxed relative to the general random-coefficient setting (e.g., deterministic vs. random coefficients).

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the detailed and constructive report. We agree that the existence of solutions to the general IESREJs is established only in a special case, and we have revised the manuscript to make this conditional nature explicit in the abstract, introduction, and main theorems. Below we respond point by point to the major comments.

read point-by-point responses

  1. Referee: [Main derivation and existence result (as summarized in the abstract)] The central closed-loop representation (derived via Meyer's Itô formula and completing the square) is expressed in terms of solutions to the IESREJs, but the manuscript proves existence of these equations only for a special case. This leaves the general result for random coefficients conditional on an unverified assumption, which is load-bearing for the claim of an explicit saddle-point representation.

    Authors: We agree with this assessment. The closed-loop representation is derived assuming solutions to the IESREJs exist; the paper proves existence only for a special case. In the revised version we have updated the abstract to read that the representation holds 'in terms of solutions to the IESREJs (whose existence is established for a special case)' and added a clarifying paragraph in the introduction and after Theorem 3.2 stating that the result is conditional on solvability of the IESREJs. This removes any implication of unconditional generality while preserving the derivation via Meyer's formula and completing the square. revision: yes

  2. Referee: [FBSDE characterization and IESREJ reduction] The FBSDE characterization of the open-loop saddle point under the UCC condition is standard, but the subsequent reduction to IESREJs for the constrained case requires the existence result to be constructive; without general existence, the four-step scheme failure noted in the paper is not fully overcome for the stated problem class.

    Authors: The FBSDE characterization under UCC is standard, as noted. The reduction to IESREJs supplies an explicit closed-loop form (via the feedback operators derived from the Riccati solutions) that circumvents the four-step scheme when the IESREJs are solvable. We acknowledge that this does not yield a fully constructive method for arbitrary random coefficients. The revision adds a remark after the main theorem explicitly discussing this limitation and noting that general existence remains open, thereby clarifying how far the approach overcomes the four-step obstruction. revision: partial

standing simulated objections not resolved
  • Proving existence of solutions to the general multidimensional indefinite extended stochastic Riccati equations with jumps for arbitrary random coefficients and regime switching.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper assumes the UCC condition to obtain FBSDE characterization of the open-loop saddle point, then applies Meyer's Itô formula and completing-the-square to express a closed-loop representation in terms of solutions to the newly defined IESREJs. This follows standard stochastic calculus steps without reducing the claimed representation to a fitted parameter defined inside the paper or to a self-citation whose content is unverified. Existence of IESREJ solutions is established only for a special case, but that is an applicability limitation rather than a definitional loop; the central derivation remains independent of the target result and self-contained against external stochastic control benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the uniform convexity-concavity condition as a domain assumption and on the solvability of the newly introduced IESREJs, whose existence is shown only in a special case.

axioms (1)
  • domain assumption Uniform convexity-concavity (UCC) condition holds for the game
    Invoked to guarantee open-loop solvability and the FBSDE characterization of the saddle point.
invented entities (1)
  • Indefinite extended stochastic Riccati equations with jumps (IESREJs) no independent evidence
    purpose: Provide the closed-loop representation of the open-loop saddle point when controls are cone-constrained
    New family of equations introduced in the paper; independent existence proof supplied only for a special case.

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Reference graph

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