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arxiv: 2605.25719 · v2 · pith:MJF7PU6Gnew · submitted 2026-05-25 · 🧮 math.PR

A note on convergence rate for reflected BSDEs with quadratic generators by penalization method

Guangyan Jia , Peng Luo , Mengbo Zhu This is my paper

Pith reviewed 2026-06-29 20:45 UTC · model grok-4.3

classification 🧮 math.PR
keywords reflected BSDEsquadratic generatorspenalizationconvergence rateBMO martingalesnumerical approximationEuler method
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The pith

Reflected BSDEs with quadratic generators converge to their penalized counterparts at rate 1/2.

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

This paper proves that the solutions to reflected backward stochastic differential equations with quadratic generators and the solutions to their penalized versions differ by an amount that shrinks like the square root of the penalty parameter. The proof relies on estimates from BMO martingales applied to the difference processes. A reader might care because the explicit rate gives a concrete way to balance penalty size against approximation error when solving these equations numerically. The result is then used to justify an Euler polygonal line scheme for reflected BSDEs that have only sub-quadratic generators.

Core claim

Using techniques of BMO martingales, we prove the convergence rate is at order 1/2 as a function of the penalty parameter between reflected BSDEs with quadratic generators and their penalized BSDEs. The result is applied to study numerical approximation of reflected BSDEs with sub-quadratic generators by the Euler's polygonal line method.

What carries the argument

BMO martingale techniques controlling the difference process between the reflected and penalized equations.

If this is right

  • The error between reflected and penalized solutions is bounded by C times the square root of the penalty parameter.
  • This bound holds for quadratic generators under standard integrability conditions.
  • The rate extends the applicability of penalization to numerical methods for reflected BSDEs.
  • The same penalization approach works for sub-quadratic generators in the Euler scheme context.

Where Pith is reading between the lines

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

  • Whether the 1/2 rate is optimal could be checked by constructing an example where the error decays exactly like sqrt(ε).
  • Similar BMO estimates might yield rates for other approximation schemes such as time-discretization of the reflected equations.
  • Connections to optimal stopping problems could follow since reflected BSDEs often represent solutions to those.

Load-bearing premise

The generators and terminal conditions admit BMO-martingale estimates that control the difference between the reflected equation and the penalized equation.

What would settle it

Finding a quadratic generator and terminal condition where the L^p norm of the difference between reflected and penalized solutions fails to be O(ε^{1/2}) for small penalty parameter ε.

read the original abstract

In this paper, we study the convergence rate between reflected backward stochastic differential equations with quadratic generators and their penalized BSDEs. Using techniques of BMO martingales, we prove the convergence rate is at order $\frac{1}{2}$ as a function of the penalty parameter. Finally, the result is applied to study numerical approximation of reflected BSDEs with sub-quadratic generators by the Euler's polygonal line method.

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 / 1 minor

Summary. The paper studies reflected BSDEs with quadratic generators and their penalized approximations. It claims to prove, via BMO-martingale techniques, that the convergence rate between the reflected solution and the penalized solution is of order 1/2 in the penalty parameter. The result is then applied to obtain a numerical approximation result for reflected BSDEs with sub-quadratic generators using the Euler polygonal-line method.

Significance. If the rate holds under the stated conditions, the quantitative error bound would be useful for justifying penalization as a practical approximation tool and for error analysis in numerical schemes for reflected BSDEs. The application to Euler discretization adds a concrete consequence. The abstract, however, supplies neither the precise assumption list nor any derivation steps, so the significance cannot be fully assessed from the given information.

major comments (1)
  1. [Proof of the main convergence-rate result (as described in the abstract)] The central claim (abstract) asserts that BMO-martingale estimates applied to the difference process between the reflected and penalized solutions close at rate 1/2 uniformly in the penalty parameter n. The penalized driver contains the term n(Y^n - L)^- whose size grows linearly with n. Without an explicit a priori bound showing that the BMO norm of this difference driver (or its quadratic variation) remains controlled independently of n, the standard exponential-martingale or quadratic-variation comparison may produce a Gronwall factor that diverges with n, preventing the claimed rate. The manuscript must supply the missing uniform estimate or show why it is unnecessary.
minor comments (1)
  1. [Abstract] The abstract states the result but lists neither the precise integrability/growth assumptions on the generator and terminal condition nor the definition of the penalty term, making it impossible to verify applicability of the BMO estimates from the abstract alone.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and insightful comments on our manuscript. We address the major comment below.

read point-by-point responses

  1. Referee: [Proof of the main convergence-rate result (as described in the abstract)] The central claim (abstract) asserts that BMO-martingale estimates applied to the difference process between the reflected and penalized solutions close at rate 1/2 uniformly in the penalty parameter n. The penalized driver contains the term n(Y^n - L)^- whose size grows linearly with n. Without an explicit a priori bound showing that the BMO norm of this difference driver (or its quadratic variation) remains controlled independently of n, the standard exponential-martingale or quadratic-variation comparison may produce a Gronwall factor that diverges with n, preventing the claimed rate. The manuscript must supply the missing uniform estimate or show why it is unnecessary.

    Authors: We thank the referee for highlighting this potential issue in the application of BMO estimates. The current proof applies BMO-martingale techniques to the difference process but does not isolate an explicit uniform-in-n bound on the BMO norm of the penalized driver term as a separate step. We agree that making this control explicit would strengthen the argument and prevent any concern about n-dependent factors in the estimates. We will revise the manuscript by adding a preliminary lemma that establishes the required uniform BMO bound on the difference driver, derived from the quadratic growth condition, comparison principles, and a priori L^2 estimates on the solutions that hold independently of n. This addition will be incorporated into Section 3 without changing the main result or assumptions. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation relies on standard BMO estimates

full rationale

The provided abstract and context describe a proof of an O(1/sqrt(n)) convergence rate between reflected BSDEs and penalized approximations via BMO-martingale techniques. No equations, fitted parameters, self-citations, or ansatzes are visible that reduce the claimed rate to the inputs by construction. BMO estimates are external, independently verifiable tools in stochastic analysis and do not depend on the paper's own fitted quantities or prior self-referential results. The central claim therefore remains non-circular.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review; the proof is asserted to rest on standard BMO-martingale estimates and quadratic-growth conditions typical in reflected BSDE theory.

axioms (2)
  • domain assumption Quadratic generators admit unique solutions to the reflected BSDE and the penalized BSDE under the stated integrability.
    Required for the difference process to be well-defined before BMO estimates are applied.
  • domain assumption BMO-martingale inequalities apply directly to the error process arising from penalization.
    Central technical step invoked to obtain the 1/2 rate.

pith-pipeline@v0.9.1-grok · 5588 in / 1258 out tokens · 33782 ms · 2026-06-29T20:45:33.442689+00:00 · methodology

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Reference graph

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