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arxiv: 2604.23656 · v1 · submitted 2026-04-26 · 🧮 math.PR

Doubly Reflected Backward SDEs Driven by G-Brownian Motion with Quadratic Generator

Hanwu Li , Peng Luo , Mengbo Zhu This is my paper

Pith reviewed 2026-05-08 05:28 UTC · model grok-4.3

classification 🧮 math.PR
keywords doubly reflected G-BSDEsquadratic generatorG-Brownian motionexistence and uniquenesspenalization approximationG-BMO martingalesdouble-obstacle PDEs
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The pith

Doubly reflected G-BSDEs with quadratic generators have unique solutions when the upper obstacle is almost a generalized G-Itô process.

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

The paper establishes existence and uniqueness for doubly reflected backward SDEs driven by G-Brownian motion with a quadratic generator. The key condition is that the upper obstacle is almost a generalized G-Itô's process. Under this assumption the solution is the monotone limit of solutions to penalized reflected G-BSDEs that involve only a lower obstacle. The construction relies on G-BMO martingale theory and the G-Girsanov theorem. This monotone approximation is then used to relate the stochastic equations to fully nonlinear PDEs with two obstacles.

Core claim

When the upper obstacle is almost a generalized G-Itô's process, the doubly reflected G-BSDE with quadratic generator admits a unique solution. The solution is obtained as the monotone limit of solutions to a family of penalized reflected G-BSDEs with a lower obstacle only.

What carries the argument

Penalization scheme for the upper obstacle, combined with G-BMO martingale estimates and G-Girsanov change of measure to handle quadratic growth.

If this is right

  • The solution is the monotone limit of solutions to penalized reflected G-BSDEs with only a lower obstacle.
  • The penalization procedure connects doubly reflected G-BSDEs to fully nonlinear PDEs with double obstacles.
  • The G-BMO and G-Girsanov tools extend existence results to the double-reflection case with quadratic terms.

Where Pith is reading between the lines

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

  • This penalization limit could support numerical approximation schemes for the associated double-obstacle PDEs under model uncertainty.
  • The approach may generalize to other reflected G-BSDE problems arising in robust control.
  • Comparison principles for the corresponding nonlinear PDEs become accessible through the same estimates.

Load-bearing premise

The upper obstacle must be almost a generalized G-Itô's process in order for G-BMO theory to control the quadratic growth term.

What would settle it

An explicit counter-example of a quadratic doubly reflected G-BSDE whose upper obstacle fails to be almost a generalized G-Itô process, yet which still possesses a solution, would disprove the claim.

read the original abstract

In this paper, we study the doubly reflected backward stochastic differential equations driven by $G$-Brownian motion ($G$-BSDEs for short) when the generator has quadratic growth in the $z$-component. Based on the theory of $G$-BMO martingale and $G$-Girsanov theorem, we establish the existence and uniqueness result when the upper obstacle is almost a generalized $G$-It\^{o}'s process. Moreover, the solution can be approximated monotonically by the solutions to a family of penalized reflected $G$-BSDEs with a lower obstacle, which plays an important role to establish the relation between doubly reflected $G$-BSDEs and fully nonlinear partial differential equations with double obstacles.

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

1 major / 2 minor

Summary. The paper establishes existence and uniqueness for solutions to doubly reflected G-BSDEs with quadratic growth in the z-component, under the standing assumption that the upper obstacle is almost a generalized G-Itô process. The argument proceeds by solving a family of penalized single-reflected G-BSDEs (lower obstacle only) and passing to the monotone limit, employing G-BMO martingale estimates together with the G-Girsanov theorem to control the quadratic term. The resulting approximation is presented as a tool for linking the doubly reflected equation to fully nonlinear PDEs with double obstacles.

Significance. If the stated existence/uniqueness result holds, the work supplies a technically consistent extension of the G-framework from single-reflection and linear-growth cases to the doubly reflected quadratic setting. The monotone penalization scheme is a standard and useful device that directly supports the PDE connection claimed in the abstract. The explicit restriction on the upper obstacle is stated up front, avoiding hidden assumptions.

major comments (1)
  1. [§3] §3 (main existence theorem): the application of the G-Girsanov theorem to absorb the quadratic generator term requires verification that the G-BMO norm remains controlled after the measure change; the manuscript should supply the explicit estimate showing that the transformed process stays in G-BMO under the given obstacle hypothesis.
minor comments (2)
  1. [Abstract/Introduction] The abstract and introduction use the phrase 'almost a generalized G-Itô process' without a precise definition or reference to the exact integrability conditions; a short paragraph clarifying this notion (e.g., via the decomposition into G-Itô integral plus bounded variation term) would improve readability.
  2. [§2] Notation for the penalized processes (e.g., the penalty parameter and the reflected processes) should be introduced once in §2 and used consistently thereafter to avoid minor confusion in the limit passage.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of the manuscript and for the constructive comment on the main existence theorem. We address the point below and will incorporate the requested verification in the revised version.

read point-by-point responses

  1. Referee: [§3] §3 (main existence theorem): the application of the G-Girsanov theorem to absorb the quadratic generator term requires verification that the G-BMO norm remains controlled after the measure change; the manuscript should supply the explicit estimate showing that the transformed process stays in G-BMO under the given obstacle hypothesis.

    Authors: We agree that an explicit estimate is required to rigorously justify the G-Girsanov transformation in the proof. In the revised manuscript we will insert a dedicated lemma (placed immediately before the application of G-Girsanov) that derives the bound on the G-BMO norm of the transformed process. The argument uses the standing hypothesis that the upper obstacle is nearly a generalized G-Itô process to control the additional drift terms generated by the measure change, thereby ensuring the quadratic generator remains integrable with respect to the new measure. This addition will make the passage from the penalized equations to the doubly reflected limit fully justified. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's central result is an existence/uniqueness theorem for doubly reflected G-BSDEs with quadratic generator, obtained by solving a family of penalized single-obstacle equations and passing to the monotone limit under the explicit hypothesis that the upper obstacle is almost a generalized G-Itô process. The argument relies on G-BMO martingale estimates and the G-Girsanov theorem, which are invoked as established external tools from prior G-framework literature rather than being defined or fitted inside the present work. No step reduces the target statement to a self-defined quantity, a fitted input renamed as prediction, or a load-bearing self-citation chain; the obstacle restriction is stated upfront and the approximation technique is the standard device for double-obstacle problems. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the pre-existing G-expectation framework and associated martingale theory without introducing new free parameters or postulated entities; the contribution is the extension of existence results to the quadratic doubly-reflected setting.

axioms (2)
  • domain assumption Existence and basic properties of G-Brownian motion and G-expectation
    Invoked as the driving noise and probability framework throughout.
  • domain assumption Theory of G-BMO martingales and G-Girsanov theorem
    Used to control the quadratic generator term and change of measure.

pith-pipeline@v0.9.0 · 5419 in / 1291 out tokens · 55311 ms · 2026-05-08T05:28:00.168399+00:00 · methodology

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