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arxiv: 2604.22130 · v1 · submitted 2026-04-24 · 🧮 math.PR

Reflected Stochastic Differential Equations Driven by G-Brownian Motion with Nonlinear Constraints

Hanwu Li This is my paper

Pith reviewed 2026-05-08 10:31 UTC · model grok-4.3

classification 🧮 math.PR
keywords reflected stochastic differential equationsG-Brownian motionnonlinear constraintsSkorokhod problemPicard iterationcomparison theoremG-expectationdoubly reflected processes
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The pith

Reflected G-SDEs with nonlinear constraints admit unique solutions constructed via the Skorokhod problem and Picard iteration.

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

The paper proves existence and uniqueness for stochastic differential equations driven by G-Brownian motion that are reflected at two nonlinear boundaries. It first constructs the doubly reflected G-Brownian motion pathwise by solving the associated Skorokhod problem while remaining inside the original G-expectation space. Uniqueness then follows from a priori estimates on the coefficients, existence is shown by convergence of a Picard sequence, and a comparison theorem orders the solution against each individual constraining process. These results extend classical reflected diffusion theory to the nonlinear-expectation setting used to model Knightian uncertainty.

Core claim

With the Skorokhod problem with nonlinear constraints in hand, the doubly reflected G-Brownian motion can be built pathwise and stays in the same G-expectation space as the driving process. The reflected G-SDE then possesses a unique solution: uniqueness is obtained from an a priori estimate, existence follows from the convergence of Picard iterates, and the solution satisfies a comparison theorem with respect to each of the two nonlinear constraining processes.

What carries the argument

The Skorokhod problem with nonlinear constraints, which supplies the pathwise construction of the doubly reflected G-Brownian motion inside the G-expectation space and supplies the reflection terms for the SDE.

If this is right

  • Uniqueness of solutions to the reflected G-SDE follows from the a priori estimate once the Skorokhod problem is solved.
  • Existence is obtained by showing that the Picard iteration converges in the G-expectation space.
  • The comparison theorem orders any solution between the two nonlinear constraining processes.
  • The pathwise construction keeps the reflected process inside the same G-framework as the driving G-Brownian motion.

Where Pith is reading between the lines

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

  • The pathwise construction may simplify Monte-Carlo simulation of reflected processes under G-uncertainty.
  • The comparison theorem supplies a tool for proving monotonicity properties when constraints vary with model uncertainty.
  • Similar Skorokhod-based arguments could be tested on reflected equations driven by other nonlinear expectations.

Load-bearing premise

The Skorokhod problem with nonlinear constraints admits a unique solution inside the G-expectation space and the coefficient conditions are strong enough for the required a priori estimates to hold.

What would settle it

A concrete coefficient set for which either the Skorokhod problem has more than one solution or the Picard iterates fail to converge to a unique limit would show that existence or uniqueness fails.

read the original abstract

In this paper, we study the reflected stochastic differential equations driven by G-Brownian motion (reflected G-SDEs) with two nonlinear constraints. With the help of the Skorokhod problem with nonlinear constraints, we first study the doubly reflected G-Brownian motion, which is constructed pathwise and lies in the same G-expectation space as the G-Brownian motion. For the reflected G-SDE, the uniqueness is derived from some a priori estimate and the existence is obtained by a Picard iteration method. The comparison theorem of the solution and the individual constraining processes are provided.

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 / 1 minor

Summary. The manuscript studies reflected stochastic differential equations driven by G-Brownian motion subject to two nonlinear constraints. It first constructs the doubly reflected G-Brownian motion pathwise by solving the associated Skorokhod problem with nonlinear constraints, verifying that the resulting process remains in the same G-expectation space as the driving G-Brownian motion. For the reflected G-SDE itself, uniqueness is obtained from an a priori estimate while existence follows from a Picard iteration argument; a comparison theorem relating the solution to the individual constraining processes is also stated.

Significance. If the central claims hold, the work extends reflected SDE theory to the G-framework with nonlinear barriers, providing pathwise constructions and comparison results that could support applications in robust optimization or uncertainty modeling. The explicit use of the nonlinear Skorokhod map and the preservation of the G-expectation space are potentially valuable technical contributions.

major comments (2)
  1. [Abstract / construction of doubly reflected G-Brownian motion] The well-posedness of the Skorokhod problem with nonlinear constraints under G-expectation is the foundational step for the pathwise construction of the doubly reflected G-Brownian motion. Because G-expectation is sublinear and admits no single dominating probability measure, standard arguments for uniqueness of the reflection map may fail to carry over; the manuscript must supply a self-contained proof or a precise reference that covers the nonlinear case without hidden regularity assumptions on the constraints.
  2. [Uniqueness argument for the reflected G-SDE] The a priori estimate invoked for uniqueness of the reflected G-SDE must be shown to remain valid once the nonlinear reflection terms are included. It is unclear whether the estimate closes without additional Lipschitz or growth conditions on the constraining processes that are not stated in the abstract; if the estimate relies on the linear case, the uniqueness claim for the nonlinear setting is not yet load-bearing.
minor comments (1)
  1. The abstract would be clearer if it listed the precise assumptions on the drift, diffusion, and constraint functions (e.g., Lipschitz constants, growth bounds) under which the Picard iteration and a priori estimate are performed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and valuable suggestions. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Abstract / construction of doubly reflected G-Brownian motion] The well-posedness of the Skorokhod problem with nonlinear constraints under G-expectation is the foundational step for the pathwise construction of the doubly reflected G-Brownian motion. Because G-expectation is sublinear and admits no single dominating probability measure, standard arguments for uniqueness of the reflection map may fail to carry over; the manuscript must supply a self-contained proof or a precise reference that covers the nonlinear case without hidden regularity assumptions on the constraints.

    Authors: We agree that due to the sublinear nature of G-expectation, the well-posedness of the Skorokhod problem with nonlinear constraints requires careful treatment. In the manuscript, we construct the doubly reflected G-Brownian motion pathwise by solving the associated Skorokhod problem and verify that the resulting process remains in the G-expectation space. However, to address the referee's concern, we will include a self-contained proof of the uniqueness of the nonlinear Skorokhod map under G-expectation in the revised manuscript. This proof will explicitly handle the absence of a dominating measure by leveraging the properties of G-Brownian motion and the nonlinear constraints, without additional regularity assumptions beyond those stated. revision: yes

  2. Referee: [Uniqueness argument for the reflected G-SDE] The a priori estimate invoked for uniqueness of the reflected G-SDE must be shown to remain valid once the nonlinear reflection terms are included. It is unclear whether the estimate closes without additional Lipschitz or growth conditions on the constraining processes that are not stated in the abstract; if the estimate relies on the linear case, the uniqueness claim for the nonlinear setting is not yet load-bearing.

    Authors: The a priori estimate for uniqueness is derived using the G-Itô formula applied to the squared difference of two potential solutions, incorporating the reflection terms directly. The nonlinear constraints are accounted for through their monotonicity properties, which ensure that the cross terms contribute non-positively. This estimate does not rely on the linear case and closes under the Lipschitz and growth conditions already assumed for the coefficients and constraints in the paper. We will revise the uniqueness section to include more explicit calculations showing how the nonlinear terms are handled, making the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation builds on external Skorokhod well-posedness without self-referential reduction.

full rationale

The paper first invokes the Skorokhod problem with nonlinear constraints to construct the doubly reflected G-Brownian motion pathwise within the G-expectation space, then derives uniqueness of the reflected G-SDE via a priori estimates and existence via Picard iteration, followed by a comparison theorem. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or unverified self-citation chain by construction. The central claims remain independent of the inputs once the (external) Skorokhod foundation is granted, satisfying the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the existence and properties of G-Brownian motion and the well-posedness of the nonlinear Skorokhod problem; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption G-Brownian motion exists and generates a consistent G-expectation space in which the Skorokhod problem can be solved pathwise.
    Invoked to place the doubly reflected process in the same space as the driving noise.
  • domain assumption The nonlinear Skorokhod problem with two constraints admits a unique solution.
    Used as the foundation for constructing the reflected G-Brownian motion.

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Reference graph

Works this paper leans on

35 extracted references · 35 canonical work pages

  1. [1]

    The Skorokhod problem in a time-dependent interval

    Burdzy K, Kang W, Ramanan K. The Skorokhod problem in a time-dependent interval. Stochastic Processes and their Applications, 2009, 119: 428-452

    work page 2009

  2. [2]

    Brownian motion reflected on Brownian motion

    Burdzy K, Nualart D. Brownian motion reflected on Brownian motion. Probab. Theor. Rel. Fields, 2002, 122: 471-493

    work page 2002

  3. [3]

    Un probl` eme de r´ eflexion et ses applications au temps local et aux ´ equations diff´ erentielles stochastiques surR, cas continu

    Chaleyat-Maurel M, El Karoui N. Un probl` eme de r´ eflexion et ses applications au temps local et aux ´ equations diff´ erentielles stochastiques surR, cas continu. Expos´ es du S´ eminaire J. Az´ ema-M. Yor. Held at the Universit´ e Pierre et Marie Curie, Paris, 1976-1977, pp. 117–144. Ast´ erisque, 52, 53, Soci´ et´ e Math´ ematique de France, Paris, 1978

    work page 1976

  4. [4]

    Function spaces and capacity related to a sublinear expectation: appli- cation toG-Brownian motion paths

    Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: appli- cation toG-Brownian motion paths. Potential Anal., 2011, 34: 139-161

    work page 2011

  5. [5]

    A new approach to the Skorohod problem, and its applications

    El Karoui N, Karatzas I. A new approach to the Skorohod problem, and its applications. Stochas- tics Stochastics Rep., 1991, 34: 57-82

    work page 1991

  6. [6]

    Pathwise properties and homeomorphic flows for stochastic differential equations driven byG-Brownian motion

    Gao F. Pathwise properties and homeomorphic flows for stochastic differential equations driven byG-Brownian motion. Stochastic Process. Appl., 2009, 119: 3356–3382

    work page 2009

  7. [7]

    Local time, coupling and the passport option

    Henderson V, Hobson D. Local time, coupling and the passport option. Finance Stoch., 2000, 4: 69-80

    work page 2000

  8. [8]

    Quasi-continuous random variables and processes under theG- expectation framework

    Hu M, Wang F, Zheng G. Quasi-continuous random variables and processes under theG- expectation framework. Stochastic Process. Appl., 2016, 126(8): 2367-2387

    work page 2016

  9. [9]

    On reflection with two-sided jumps

    Jarni I, Ouknine Y. On reflection with two-sided jumps. Journal of Theoretical Probability, 2021, 34: 1811-1830

    work page 2021

  10. [10]

    An explicit formula for the Skorokhod map on [0, a]

    Kruk L, Lehoczky J, Ramanan K, Shreve S. An explicit formula for the Skorokhod map on [0, a]. Annals of Probability, 2007, 35(5): 1740-1768

    work page 2007

  11. [11]

    The Skorokhod problem with two nonlinear constraints

    Li H. The Skorokhod problem with two nonlinear constraints. Probability and Mathematical Statistics, 2023, 43(2): 207-239

    work page 2023

  12. [12]

    Stochastic differential equations driven byG-Brownian motion with mean reflec- tions, 2023, arXiv: 2306.08931v3

    Li H, Ning N. Stochastic differential equations driven byG-Brownian motion with mean reflec- tions, 2023, arXiv: 2306.08931v3

    work page arXiv 2023

  13. [13]

    Doubly reflected backward SDEs driven by G-Brownian motions and fully non- linear PDEs with double obstacles

    Li H, Ning N. Doubly reflected backward SDEs driven by G-Brownian motions and fully non- linear PDEs with double obstacles. Stochastics and Partial Differential Equations: Analysis and Computations, 2025, 13: 1279-1318

    work page 2025

  14. [14]

    Supermartingale decomposition theorem underG-expectation

    Li H, Peng S, Song Y. Supermartingale decomposition theorem underG-expectation. Electron. J. Probab., 2018, 23: 1-20. 23

    work page 2018

  15. [15]

    Backward stochastic differential equations driven byG-Brownian motion with double reflections

    Li H, Song Y. Backward stochastic differential equations driven byG-Brownian motion with double reflections. Journal of Theoretical Probability, 2021, 34: 2285-2314

    work page 2021

  16. [16]

    Stopping times and related Itˆ o’s calculus withG-Brownian motion

    Li X, Peng S. Stopping times and related Itˆ o’s calculus withG-Brownian motion. Stochastic Process. Appl., 2011, 121: 1492-1508

    work page 2011

  17. [17]

    Stochastic differential equations driven byG-Brownian motion with reflecting boundary conditions

    Lin Y. Stochastic differential equations driven byG-Brownian motion with reflecting boundary conditions. Electron. J. Probab., 2013, 18: 1-24

    work page 2013

  18. [18]

    ´Equations diff´ erentielles stochastiques sous les esp´ erances math´ ematiques non-lin´ eaires et applications

    Lin Y. ´Equations diff´ erentielles stochastiques sous les esp´ erances math´ ematiques non-lin´ eaires et applications. PhD thesis, 2013, Universit´ e de Rennes

    work page 2013

  19. [19]

    Reflected stochastic differential equations driven byG-Brownian motion in non-convex domains

    Lin Y, Soumana Hima A. Reflected stochastic differential equations driven byG-Brownian motion in non-convex domains. Stochastics and Dynamics, 2019, 19(3): 1950025

    work page 2019

  20. [20]

    Stochastic differential equations with reflecting boundary conditions

    Lions P L, Sznitman A L. Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math., 1984, 37: 511–537

    work page 1984

  21. [21]

    Reflected stochastic differential equations driven byG-Brownian motion with nonlinear resistance

    Luo P. Reflected stochastic differential equations driven byG-Brownian motion with nonlinear resistance. Front. Math. China, 2016, 11(1): 123–140

    work page 2016

  22. [22]

    Stochastic differential equations driven byG-Brownian motion and ordinary differential equations

    Luo P, Wang F. Stochastic differential equations driven byG-Brownian motion and ordinary differential equations. Stochastic Process. Appl., 2014, 124(11): 3869-3885

    work page 2014

  23. [23]

    Strong approximations for time-dependent queues

    Mandelbaum A, Massey W. Strong approximations for time-dependent queues. Math. Oper. Res., 1995, 20: 33-63

    work page 1995

  24. [24]

    Nonlinear Expectations and Stochastic Calculus Under Uncertainty: With Robust CLT and G-Brownian Motion

    Peng S. Nonlinear Expectations and Stochastic Calculus Under Uncertainty: With Robust CLT and G-Brownian Motion. Probability Theory and Stochastic Modelling 95, Springer, 2019

    work page 2019

  25. [25]

    Stochastic equations for diffusions in a bounded region

    Skorokhod A.V. Stochastic equations for diffusions in a bounded region. Theory Probab. Appl., 1961, 6: 264-274

    work page 1961

  26. [26]

    Explicit representation of the Skorokhod map with time dependent boundaries

    Slaby M. Explicit representation of the Skorokhod map with time dependent boundaries. Prob- ability and Mathematical Statistics, 2010, 30: 29-60

    work page 2010

  27. [27]

    An explicit representation of the extended Skorokhod map with two time-dependent boundaries

    Slaby M. An explicit representation of the extended Skorokhod map with two time-dependent boundaries. Journal of Probability and Statistics, 2010, 2010: 1-18

    work page 2010

  28. [28]

    On existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting conditions

    Slomi´ nski L. On existence, uniqueness and stability of solutions of multidimensional SDE’s with reflecting conditions. Ann. lIHP Probab. Stat., 1993, 29: 163-198

    work page 1993

  29. [29]

    On approximation of solutions of multidimensional SDEs with reflecting boundary conditions

    Slomi´ nski L. On approximation of solutions of multidimensional SDEs with reflecting boundary conditions. Stoch. Process. Appl., 1994, 50: 197-219

    work page 1994

  30. [30]

    Stochastic differential equations with jump reflection at time- dependent barriers

    Slomi´ nski L, Wojciechowski T. Stochastic differential equations with jump reflection at time- dependent barriers. Stoch. Process. Appl., 2010, 120: 1701-1721

    work page 2010

  31. [31]

    Stochastic differential equations with time-dependent reflecting barriers

    Slomi´ nski L, Wojciechowski T. Stochastic differential equations with time-dependent reflecting barriers. Stochastics, 2013, 85(1): 27-47

    work page 2013

  32. [32]

    A note on reflecting Brownian motions

    Soucaliuc F, Werner W. A note on reflecting Brownian motions. Electon. Commum. Probab., 2002, 7: 117-122

    work page 2002

  33. [33]

    Large deviation principle for reflected stochastic differential equa- tions driven byG-Brownian motion in non-convex domains

    Soumana Hima A, Dakaou I. Large deviation principle for reflected stochastic differential equa- tions driven byG-Brownian motion in non-convex domains. Statistics and Probability Letters, 2023, 193: 109707. 24

    work page 2023

  34. [34]

    Stochastic differential equations with reflecting boundary condition in convex regions

    Tanaka, H. Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J., 1979, 9: 163-177

    work page 1979

  35. [35]

    An Introduction to Stochastic Process Limits and Their Application to Queues

    Whitt W. An Introduction to Stochastic Process Limits and Their Application to Queues. Springer, New York, 2002. 25

    work page 2002