Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions

Shiqiu Zheng

Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions

T0 review · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Quadratic BSDEs and reflected BSDEs with one-sided growth in y admit solutions when generators satisfy convexity conditions.

desk verdict This paper adapts existing existence results for quadratic BSDEs to the reflected case with one-sided y-growth, then adds a uniqueness argument via θ-difference and new comparison theorems under convexity conditions. read the letter →

arxiv 2412.21172 v6 submitted 2024-12-30 math.PR

Shiqiu Zheng This is my paper

classification math.PR

keywords backwardstochasticdifferentialequationsreflectedquadraticgrowthone-sidedwell-posednessuniquenesscomparisontheorems

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper investigates the well-posedness of solutions to backward stochastic differential equations and their reflected versions. The generators satisfy a one-sided growth restriction on the variable y combined with quadratic growth in z, and solutions may be bounded or unbounded with values in the reals or positive numbers. Existence is obtained by adapting existing approaches for these growth conditions. Uniqueness is established when the generators are convex in the pair of variables or locally Lipschitz in y and convex in z, relying on a difference technique and novel comparison arguments. General comparison theorems are derived for these classes of equations.

What carries the argument

A difference technique applied to the reflected equations, which supports uniqueness under convexity assumptions on the generator.

What would settle it

A generator meeting the one-sided and quadratic growth conditions that is convex in both variables but for which the corresponding equation has either no solution or multiple solutions.

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Extended reading notes

Core claim

The paper establishes that for generators with one-sided growth restriction on y and quadratic growth in z, existence of solutions to the reflected and non-reflected equations follows from adapted standard methods, while uniqueness holds under convexity in both variables or local Lipschitz continuity in y with convexity in z. The proof of uniqueness employs a difference technique together with comparison arguments based on the reflected equations, and general comparison theorems are also proved for the equations.

Load-bearing premise

The one-sided growth restriction on y together with quadratic growth in z remains compatible with comparison arguments and the difference technique without requiring extra regularity that would invalidate existence.

Editorial extensions

If this is right

Existence and uniqueness results apply to both bounded and unbounded solutions.
The solutions can take values in the reals or the positive reals.
General comparison theorems hold for the reflected and non-reflected equations under the given growth conditions.
Uniqueness is available when the generator is convex or satisfies a local Lipschitz-convex combination.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Such well-posedness results could support analysis of related stochastic control problems with quadratic terms.
The comparison theorems might be useful for establishing monotonicity properties in applications.
Extensions to other types of stochastic equations with similar growth restrictions could be explored.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The core contribution is existence for both RBSDEs and BSDEs under one-sided linear growth in y plus quadratic growth in z, with solutions allowed to be bounded or unbounded and taking values in R or (0,∞). They pull the existence step directly from Essaky-Hassani 2011 and Bahlali et al. 2017, which is straightforward. For uniqueness they introduce a method that works when the generator is convex in (y,z) or locally Lipschitz in y and convex in z; this relies on the Briand-Hu θ-difference technique plus some comparison theorems they derive for these equations. That comparison step is the part that is actually new on the page. The setup covers both reflected and non-reflected cases, which is a modest but clean extension. The main limitation is that only the abstract is available, so it is impossible to check whether the new comparison arguments survive the one-sided growth restriction without extra regularity that would break the cited existence theorems. The abstract does not flag any obvious gaps, but the uniqueness claim rests on those comparisons holding, and that needs the full proofs. This is the sort of incremental technical note that people who already work with quadratic BSDEs in stochastic control or mathematical finance might want to cite once the details are verified. It does not open a new framework or resolve a long-standing open problem, but the claims are stated clearly enough that a referee could evaluate them in a few hours. I would send it to peer review rather than desk-reject; the existence part is borrowed but the uniqueness method deserves a proper check.

Referee Report

0 major / 0 minor

Summary. The manuscript investigates well-posedness (existence and uniqueness) for bounded and unbounded solutions of RBSDEs and BSDEs whose generators obey a one-sided growth restriction in y together with quadratic growth in z; solutions Y (and obstacles) take values in R or (0,∞). Existence is obtained by adapting the methods of Essaky-Hassani (2011) and Bahlali et al. (2017). Uniqueness is proved when the generator is convex in (y,z) or locally Lipschitz in y and convex in z, via the θ-difference technique of Briand-Hu (2008) combined with new comparison arguments for RBSDEs; general comparison theorems for such equations are also derived.

Significance. If the adaptation of the cited existence results and the new comparison arguments are valid under the stated one-sided growth, the paper would extend the quadratic-BSDE theory to a broader class of generators that arise in applications with asymmetric constraints. The general comparison theorems could be reusable tools for uniqueness proofs in related reflected and non-reflected settings.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for acknowledging the potential significance of extending quadratic BSDE/RBSDE theory to generators with one-sided growth in y. The 'uncertain' recommendation appears to stem from the need to confirm that the cited existence methods and new comparison arguments remain valid under the one-sided growth condition. We address this below in the absence of enumerated major comments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity identified

full rationale

The abstract attributes existence directly to independent external methods in Essaky and Hassani (2011) and Bahlali et al. (2017). Uniqueness relies on the external θ-difference technique from Briand and Hu (2008) together with novel comparison arguments introduced in this work under convexity or local Lipschitz conditions. No self-citations appear, no claims reduce by definition or construction to their own inputs, and no uniqueness theorems or ansatzes are imported from the present author's prior work. The derivation chain is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The paper rests on standard stochastic analysis background (Brownian filtration, progressive measurability) and the applicability of three cited existence results; no new free parameters, invented entities, or ad-hoc axioms are introduced beyond the growth and convexity hypotheses stated in the abstract.

assumptions (1)

standard math The underlying probability space supports a Brownian motion and the usual filtration satisfying the usual conditions.
Implicit in any BSDE/RBSDE setup; required for the definition of the processes.

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Cite this review

Pith. "Pith review of Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions." pith.science (2026). https://pith.science/paper/2412.21172

@misc{pith2026241221172,
  author       = {Pith},
  title        = {Pith review of: Well-posedness of quadratic RBSDEs and BSDEs with one-sided growth restrictions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2412.21172}},
  note         = {Machine review of arXiv:2412.21172}
}

abstract

In this paper, we investigate the well-posedness of bounded and unbounded solutions for reflected backward stochastic differential equations (RBSDEs) and backward stochastic differential equations (BSDEs). The generators of these equations satisfy a one-sided growth restriction on the variable $y$ and have a general quadratic growth in the variable $z$. The solutions $Y_t$ (and the obstacles for RBSDEs) take values in either $\mathbf{R}$ or $(0, \infty)$. We obtain the existence of solutions primarily by using the methods from Essaky and Hassani (2011) and Bahlali et al. (2017). For the uniqueness of solutions, we provide a method applicable when the generators are convex in $(y,z)$ or are (locally) Lipschitz in $y$ and convex in $z$. Our method relies on the $\theta$-difference technique introduced by Briand and Hu (2008), and some novel comparison arguments based on RBSDEs. We also establish some general comparison theorems for such RBSDEs and BSDEs.

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Reviewed May 23, 2026 · model on record in the stance chip above.