arxiv: 2605.12926 · v1 · submitted 2026-05-13 · 🧮 math.OC

Recognition: unknown

Infinite-Horizon Non-Autonomous Zero-Sum Stochastic Recursive Differential Games and HJBI Equations

Qingmeng Wei, Sheng Huang

Pith reviewed 2026-05-14 18:25 UTC · model grok-4.3

classification 🧮 math.OC

keywords stochastic recursive differential gamesHJBI equationsviscosity solutionsbackward stochastic differential equationsinfinite horizonnon-autonomous systemszero-sum games

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The pith

Lower and upper value functions in infinite-horizon non-autonomous zero-sum stochastic recursive games are deterministic and uniquely solve corresponding non-autonomous HJBI equations.

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

This paper studies infinite-horizon stochastic recursive differential games between two players with zero-sum payoffs defined recursively through backward stochastic differential equations. It first proves well-posedness and stability for BSDEs with time-dependent discount factors and possibly unbounded terminal times, allowing the generator to grow non-uniformly at the origin under an integrable bound. Using finite-horizon approximations, stability estimates, and viscosity solution theory, the authors show that the lower and upper value functions are deterministic functions that serve as the unique bounded viscosity solutions to their respective time-dependent Hamilton-Jacobi-Bellman-Isaacs equations. The approach also recovers the time-homogeneous case as a special instance by using uniqueness of the non-autonomous equation rather than a probabilistic shift.

Core claim

Both the lower and upper value functions are deterministic and are the unique bounded viscosity solutions of their corresponding non-autonomous HJBI equations. In the time-homogeneous case, the value functions are independent of the initial time and solve the stationary HJBI equations in the viscosity sense.

What carries the argument

Finite horizon approximation together with BSDE stability estimates and viscosity solution arguments applied to non-autonomous HJBI equations.

If this is right

The lower and upper value functions coincide when a saddle point exists.
The time-homogeneous case follows directly from uniqueness without needing time-shift arguments.
Games with explicit time dependence in drift, diffusion, generator, and discount factor are covered.
Bounded viscosity solutions are guaranteed under the given growth conditions on the generator.

Where Pith is reading between the lines

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

This framework could support numerical approximation schemes based on finite-horizon truncations for computing infinite-horizon values.
Similar techniques might apply to nonzero-sum games or games with different payoff structures.
The deterministic nature of the values suggests that the game can be reduced to a deterministic control problem in some cases.

Load-bearing premise

The generator of the BSDE satisfies a bound |f(t,0,0)| ≤ β1(t) + β2 where β1 is integrable over infinite horizon, which is needed for the BSDE to be well-posed on the infinite time interval.

What would settle it

A counterexample where the value function depends on the starting time or fails to satisfy the HJBI equation in the viscosity sense under the stated conditions on f.

read the original abstract

In this paper, we study an infinite horizon non-autonomous stochastic recursive differential game. To this end, we first establish well-posedness and stability results for BSDEs with a time-dependent discount factor and a possibly unbounded random terminal time. The generator $f$ is allowed to be non-uniformly bounded at the origin, namely, $|f(t,0,0)|\les \beta_1(t)+\beta_2,$ $t\in[0,\infty),$ $\dbP\text{-a.s.},$ with $\beta_1\in L^1(0,\infty)$ and $\beta_2\ges0$. We then formulate a two-person zero-sum stochastic recursive differential game on the infinite horizon, where the drift, diffusion, generator and discount factor may depend explicitly on time. The lower and upper value functions are defined through Elliott--Kalton nonanticipative strategies and BSDE recursive payoffs. By finite horizon approximation, BSDE stability estimates and viscosity solution arguments, we prove that both the lower and upper value functions are deterministic and are the unique bounded viscosity solutions of their corresponding non-autonomous HJBI equations. Finally, the time-homogeneous case is recovered as a special case. Using the uniqueness of the non-autonomous HJBI equation, rather than a probabilistic shift argument, we show from the PDE viewpoint that the value functions of the autonomous system are independent of the initial time and solve the corresponding stationary HJBI equations in the viscosity sense.

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.

Desk Editor's Note private letter to a colleague

This paper extends the theory of stochastic recursive differential games to the non-autonomous infinite-horizon setting with a clean PDE-based recovery of the autonomous case.

read the letter

The main thing to know about this paper is that it handles infinite-horizon non-autonomous zero-sum stochastic recursive differential games, proving that the lower and upper value functions are deterministic and the unique bounded viscosity solutions to their non-autonomous HJBI equations. They do this by first establishing well-posedness and stability for BSDEs with time-dependent discount factors and possibly unbounded random terminal times. The generator satisfies a growth condition where |f(t,0,0)| is bounded by an L1 function plus a constant, which allows non-uniform boundedness at the origin. Using Elliott-Kalton strategies, they define the game values via BSDE payoffs. Finite-horizon approximations plus stability estimates let them pass to the infinite-horizon limit, after which viscosity solution arguments give the HJBI characterization. They recover the autonomous case by using uniqueness of the non-autonomous equation to show time-independence from the PDE side. This is a straightforward extension that puts together existing techniques in a new setting. The BSDE well-posedness under that specific bound is the foundation, and it seems to hold up. The choice to use PDE uniqueness for the autonomous recovery avoids potential issues with probabilistic shifts in the non-autonomous context. The soft spots are limited. The integrability of beta1 is crucial for the tail control, and it might be possible to weaken it, but the paper doesn't claim it's optimal. Since it leans on standard arguments, the novelty is incremental rather than revolutionary, but that's fine for this subfield. This is for researchers in stochastic control and differential games who need to deal with explicit time dependence over infinite horizons. It provides a theoretical basis that could be useful for modeling in economics or engineering with changing parameters. The work shows clear thinking and honest use of the literature. I would recommend sending it for peer review. The central claims look solid and worth a detailed check by specialists.

Referee Report

2 major / 2 minor

Summary. The paper studies infinite-horizon non-autonomous zero-sum stochastic recursive differential games. It first establishes well-posedness and stability results for BSDEs with time-dependent discount factor and possibly unbounded random terminal time, under the generator growth condition |f(t,0,0)| ≤ β1(t) + β2 with β1 ∈ L¹(0,∞). The game is formulated using Elliott-Kalton nonanticipative strategies and recursive BSDE payoffs with time-dependent coefficients. By finite-horizon approximation, BSDE stability estimates, and viscosity solution arguments, the lower and upper value functions are shown to be deterministic and the unique bounded viscosity solutions of the corresponding non-autonomous HJBI equations. The autonomous case is recovered as a special case via PDE uniqueness rather than probabilistic time-shift.

Significance. If the central claims hold, the work provides a rigorous PDE characterization for non-autonomous infinite-horizon stochastic recursive games, extending finite-horizon and autonomous results to settings with explicit time dependence in drift, diffusion, generator, and discount. The relaxed growth condition on f broadens applicability, while the use of viscosity uniqueness to recover the stationary HJBI equations offers a clean PDE-based approach. This framework could support applications in time-varying stochastic control problems in economics and finance.

major comments (2)

[BSDE well-posedness section] BSDE well-posedness section: The stability estimates for BSDEs with random terminal time and time-dependent discount rely on integrability of β1 to control tail contributions. The manuscript should provide an explicit form of the stability inequality (including dependence on the discount factor) to confirm that the estimates remain uniform as the horizon tends to infinity, which is load-bearing for the subsequent finite-horizon approximation argument.
[HJBI equations section] HJBI equations section: In the viscosity solution argument for the infinite-horizon HJBI, the passage to the limit from the finite-horizon approximations must verify that the time-dependent terms converge without introducing uncontrolled error terms in the test-function inequalities; a brief expansion of this step would strengthen the uniqueness claim.

minor comments (2)

[Game formulation] Notation for the lower and upper value functions should be defined at the start of the game formulation and used consistently thereafter.
[Viscosity comparison] A short remark on how the boundedness of viscosity solutions is preserved under the given growth condition would improve readability in the comparison principle subsection.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and have revised the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [BSDE well-posedness section] BSDE well-posedness section: The stability estimates for BSDEs with random terminal time and time-dependent discount rely on integrability of β1 to control tail contributions. The manuscript should provide an explicit form of the stability inequality (including dependence on the discount factor) to confirm that the estimates remain uniform as the horizon tends to infinity, which is load-bearing for the subsequent finite-horizon approximation argument.

Authors: We thank the referee for highlighting this point. In the revised manuscript we have added an explicit statement of the stability inequality (now displayed as (3.8) in the BSDE section). The estimate reads E[sup_{t≤τ} |Y_t - Y'_t|^2 + ∫_t^τ |Z_s - Z'_s|^2 ds] ≤ C E[∫_t^∞ e^{-∫_t^s δ(r) dr} |f(s,Y_s,Z_s) - f'(s,Y'_s,Z'_s)|^2 ds], where the constant C depends on the discount factor δ but remains independent of the (possibly random) terminal time τ thanks to the integrability of β1. This uniformity is now stated clearly and used to justify the passage to the infinite-horizon limit in the subsequent approximation argument. revision: yes
Referee: [HJBI equations section] HJBI equations section: In the viscosity solution argument for the infinite-horizon HJBI, the passage to the limit from the finite-horizon approximations must verify that the time-dependent terms converge without introducing uncontrolled error terms in the test-function inequalities; a brief expansion of this step would strengthen the uniqueness claim.

Authors: We agree that a more detailed expansion improves clarity. In the revised proof of the viscosity-solution property we have inserted an additional paragraph that explicitly tracks the convergence of the time-dependent coefficients. Using the uniform BSDE stability estimates already established, we show that the difference between the finite-horizon test-function inequalities and their infinite-horizon counterparts tends to zero; the error terms arising from the explicit time dependence are controlled by the tail integrability of β1 and therefore vanish in the limit, yielding the desired viscosity inequality without residual terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rest on external BSDE and viscosity theory

full rationale

The paper first proves well-posedness and stability for BSDEs with time-dependent discount and random terminal time under the stated integrability condition on |f(t,0,0)|. It then defines game values via Elliott-Kalton strategies and BSDE payoffs, approximates by finite horizon, and invokes standard viscosity comparison to conclude that the deterministic value functions are the unique bounded viscosity solutions of the non-autonomous HJBI equations. The autonomous case is recovered by applying the same uniqueness result rather than a probabilistic time-shift. None of these steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; all rest on independent, externally established analytic tools. Hence the central claims retain independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on preliminary well-posedness results for BSDEs with time-dependent discount and standard assumptions from stochastic control theory; no free parameters or new entities are introduced.

axioms (2)

domain assumption Well-posedness and stability of BSDEs with time-dependent discount factor and possibly unbounded random terminal time under the given generator bound
Established as a first step in the paper to support the game payoffs.
standard math Standard Lipschitz or linear growth conditions on drift, diffusion, and generator coefficients for the underlying SDEs and BSDEs
Invoked implicitly for existence of solutions in the stochastic game setup.

pith-pipeline@v0.9.0 · 5567 in / 1410 out tokens · 48346 ms · 2026-05-14T18:25:45.813271+00:00 · methodology

discussion (0)

Reference graph

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