Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem

Gechun Liang; Lianzi Jiang; Mingshang Hu

arxiv: 2506.18374 · v2 · submitted 2025-06-23 · 🧮 math.PR · cs.NA· math.NA

Probabilistic approximation of fully nonlinear second-order PIDEs with convergence rates for the universal robust limit theorem

Lianzi Jiang , Mingshang Hu , Gechun Liang This is my paper

Pith reviewed 2026-05-19 08:21 UTC · model grok-4.3

classification 🧮 math.PR cs.NAmath.NA

keywords probabilistic approximationfully nonlinear PIDEsviscosity solutionsrobust limit theoremssublinear expectationsconvergence ratesBerry-Esseen boundsLevy processes

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The pith

A probabilistic approximation scheme for nonlinear PIDEs yields explicit convergence rates for the universal robust limit theorem.

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

The paper develops a recursive piecewise-constant approximation for the viscosity solution of fully nonlinear second-order partial integro-differential equations that model nonlinear Levy processes under uncertainty. It provides explicit error estimates for this approximation. These estimates are then used to derive quantitative rates of convergence in the universal robust limit theorem under sublinear expectations. This unifies several existing limit theorems in robust probability with explicit bounds, which matters for applications where probabilities are not precisely known.

Core claim

The central claim is that a recursive, piecewise-constant probabilistic approximation of the viscosity solution to the PIDE can be constructed, with explicit error estimates. As a key application, this yields quantitative convergence rates for the universal robust limit theorem under sublinear expectations, providing a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem, together with explicit Berry-Esseen-type bounds.

What carries the argument

The recursive piecewise-constant approximation scheme for the viscosity solution of the fully nonlinear second-order PIDE featuring a supremum over a family of alpha-stable Levy measures.

If this is right

Unified treatment of Peng's robust central limit theorem and law of large numbers under the same framework.
Extension to the alpha-stable limit theorem of Bayraktar and Munk with quantitative bounds.
Explicit Berry-Esseen-type bounds for the robust limit theorems.
Handling of possibly degenerate diffusion coefficients and non-separable uncertainty sets in the approximation.

Where Pith is reading between the lines

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

The method could be adapted to derive rates for other limit theorems in sublinear expectation settings.
Practical computations of robust expectations in finance or risk management could use this approximation for numerical accuracy control.
Testing the scheme on specific alpha-stable cases would verify the rates for jump processes.

Load-bearing premise

The viscosity solution to the fully nonlinear second-order PIDE exists and is unique for the class of equations with supremum over alpha-stable Levy measures.

What would settle it

Computing the approximation error for a simple PIDE where an exact solution is known and checking if the error matches the derived rate.

read the original abstract

This paper develops a probabilistic approximation scheme for a class of nonstandard, fully nonlinear second-order partial integro-differential equations (PIDEs) associated with nonlinear Levy processes under Peng's G-expectation framework. The PIDE features a supremum over a family of alpha-stable Levy measures, possibly degenerate diffusion coefficients, and a non-separable uncertainty set, which places it outside the scope of existing numerical theories for PIDEs. We construct a recursive, piecewise-constant approximation of the viscosity solution and establish explicit error estimates for the scheme. As a key application, our results yield quantitative convergence rates for the universal robust limit theorem under sublinear expectations. This provides a unified treatment of Peng's robust central limit theorem and law of large numbers, as well as the alpha-stable limit theorem of Bayraktar and Munk, together with explicit Berry-Esseen-type bounds.

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

The paper builds a recursive probabilistic scheme for these nonstandard PIDEs with explicit error estimates and uses it to quantify rates in the universal robust limit theorem, but the viscosity comparison under degeneracy and non-separability is the part that needs checking.

read the letter

The one or two things to know are that the authors give a piecewise-constant recursive approximation to the viscosity solution of a fully nonlinear second-order PIDE that includes a supremum over alpha-stable Levy measures, possibly zero diffusion, and non-separable uncertainty, then turn the error bounds into quantitative rates for the universal robust limit theorem under sublinear expectations. This unifies Peng's robust CLT and LLN with the alpha-stable result of Bayraktar and Munk and adds explicit Berry-Esseen-type bounds. The combination sits outside prior numerical theories for PIDEs, so the scheme itself is the new piece. It does well by staying inside the established G-expectation framework without introducing circular definitions and by delivering concrete rates where only qualitative limits existed. The low circularity burden is a plus. The soft spot is the reliance on existence, uniqueness, and the comparison principle for the viscosity solution when diffusion coefficients can vanish and the uncertainty set is non-separable. Standard viscosity arguments often need uniform ellipticity or separability to build test functions and apply the maximum principle; if the paper does not supply a fresh argument for measurable selection of the optimizing measure or for uniqueness in the degenerate case, the claimed rates rest on an assumption that may not hold. The abstract gives no derivation details on this point, so the soundness is not yet clear. The stress-test note raises a fair concern here rather than a manufactured one. This paper is for people working on robust probability, sublinear expectations, or numerical methods for nonlinear PIDEs. A reader who wants quantitative versions of robust limit theorems would get direct value from the application section. It deserves a serious referee because the quantitative rates address a real gap even if revisions on the technical assumptions are likely. I would send it to peer review with instructions to focus on the viscosity theory and the handling of degeneracy.

Referee Report

2 major / 2 minor

Summary. The paper constructs a recursive piecewise-constant probabilistic approximation scheme for viscosity solutions of a class of fully nonlinear second-order PIDEs arising from nonlinear Lévy processes under Peng's G-expectation. The PIDEs involve a supremum over families of α-stable Lévy measures, possibly degenerate diffusion coefficients, and non-separable uncertainty sets. Explicit error estimates are derived for the scheme. As an application, the results yield quantitative convergence rates (including Berry-Esseen-type bounds) for the universal robust limit theorem, providing a unified treatment of Peng's robust CLT and LLN as well as the α-stable limit theorem of Bayraktar and Munk.

Significance. If the error estimates and convergence to the correct viscosity solution are rigorously justified, the work would supply the first explicit quantitative rates for robust limit theorems under sublinear expectations. This unifies and strengthens several existing qualitative limit theorems in the G-framework with concrete rates, which is a meaningful advance for nonlinear probability.

major comments (2)

[Abstract and PIDE setup] The central claim that the recursive approximation converges at explicit rates to the viscosity solution (and thereby yields the quantitative robust limit rates) rests on the existence, uniqueness, and comparison principle for the fully nonlinear PIDE with non-separable uncertainty and degenerate diffusion. The manuscript provides no derivation or citation verifying that the comparison principle holds in this setting (see abstract, paragraph 1, and the PIDE formulation). Standard viscosity theory for PIDEs typically requires separability or uniform ellipticity to construct test functions and apply the maximum principle; without explicit handling of measurable selection of the optimizing measure or degeneracy, the dynamic programming principle and error bounds cannot be guaranteed.
[Approximation scheme and error estimates] The explicit error estimates for the piecewise-constant scheme are asserted without displayed derivation steps, verification of the viscosity-solution assumption, or discussion of how degeneracy or non-separability is handled (abstract). These steps are load-bearing for the claimed Berry-Esseen-type bounds in the robust limit theorem application.

minor comments (2)

[Introduction] Notation for the family of α-stable Lévy measures and the non-separable uncertainty set should be introduced with explicit measurability conditions to facilitate reading.
[Abstract] The abstract mentions unification with Peng and Bayraktar-Munk but does not indicate which prior results are recovered as special cases or how the new rates reduce to them.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify areas where additional justification and detail are required to support the central claims. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and PIDE setup] The central claim that the recursive approximation converges at explicit rates to the viscosity solution (and thereby yields the quantitative robust limit rates) rests on the existence, uniqueness, and comparison principle for the fully nonlinear PIDE with non-separable uncertainty and degenerate diffusion. The manuscript provides no derivation or citation verifying that the comparison principle holds in this setting (see abstract, paragraph 1, and the PIDE formulation). Standard viscosity theory for PIDEs typically requires separability or uniform ellipticity to construct test functions and apply the maximum principle; without explicit handling of measurable selection of the optimizing measure or degeneracy, the dynamic programming principle and error bounds cannot be guaranteed.

Authors: We agree that the comparison principle for this non-separable, possibly degenerate PIDE requires explicit support. The manuscript relies on the probabilistic representation under the G-expectation to establish the dynamic programming principle, which circumvents some classical analytic requirements. In the revision we will add a dedicated paragraph (or short appendix) citing the relevant results from Peng's G-framework and related works on viscosity solutions for nonlinear Lévy processes that accommodate non-separable uncertainty sets via measurable selection. We will also sketch how the specific structure of the α-stable family and the sublinear expectation allow the maximum principle to hold despite degeneracy, thereby justifying the error bounds. revision: yes
Referee: [Approximation scheme and error estimates] The explicit error estimates for the piecewise-constant scheme are asserted without displayed derivation steps, verification of the viscosity-solution assumption, or discussion of how degeneracy or non-separability is handled (abstract). These steps are load-bearing for the claimed Berry-Esseen-type bounds in the robust limit theorem application.

Authors: The referee is correct that the derivation steps for the error estimates are not displayed in sufficient detail. The estimates follow from consistency of the recursive piecewise-constant scheme with the viscosity solution together with a Gronwall-type argument that exploits the Lipschitz continuity in the G-expectation. In the revised version we will insert a new subsection that (i) verifies that the scheme preserves the viscosity property, (ii) shows how the piecewise-constant approximation separately controls the degenerate diffusion and jump terms, and (iii) derives the explicit rate by combining the local truncation error with the global stability estimate. This will directly support the quantitative rates in the robust limit theorem application. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper constructs a new recursive piecewise-constant probabilistic approximation scheme for viscosity solutions of the specified class of fully nonlinear second-order PIDEs (featuring supremum over alpha-stable Levy measures and non-separable uncertainty) and derives explicit error estimates for this scheme directly from the scheme's properties. These estimates are then applied to obtain quantitative convergence rates for the universal robust limit theorem, providing a unified treatment of prior results. The approach assumes existence and uniqueness of the viscosity solution as a standing hypothesis and cites independent prior limit theorems (Peng's robust CLT/LLN and Bayraktar-Munk alpha-stable theorem) as external foundations rather than load-bearing self-references. No step reduces by construction to a fitted input, self-definitional relation, or ansatz smuggled via overlapping-author citation; the derivation remains self-contained against the established G-expectation framework and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, axioms, or invented entities are identifiable from the given text.

pith-pipeline@v0.9.0 · 5687 in / 1245 out tokens · 50809 ms · 2026-05-19T08:21:51.143935+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

40 extracted references · 40 canonical work pages

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[1] [1]

Barles and E

G. Barles and E. R. Jakobsen, On the convergence rate of approximation schemes for Hamilton-Jacobi- Bellman equations, M2AN Math. Model. Numer. Anal., 36 (2002), pp. 33-54

work page 2002

[2] [2]

Barles and E

G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for Hamilton-Jacobi- Bellman equations, SIAM J. Numer. Anal., 43 (2005), pp. 540-558

work page 2005

[3] [3]

Barles and E

G. Barles and E. R. Jakobsen, Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations. Mathematics of Computation, 76 (2007), pp. 1861-1893

work page 2007

[4] [4]

Barles and P

G. Barles and P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Analysis, 4 (1991), pp. 271-283. 23

work page 1991

[5] [5]

Bayraktar and A

E. Bayraktar and A. Munk, An α-stable limit theorem under sublinear expectation, Bernoulli, 22 (2016), pp. 2548-2578

work page 2016

[6] [6]

I. H. Biswas, I. Chowdhury, and E. R. Jakobsen, On the rate of convergence for monotone numerical schemes for nonlocal Isaacs equations, SIAM J. Numer. Anal., 57 (2019), pp. 799-827

work page 2019

[7] [7]

I. H. Biswas, E. R. Jakobsen, and K. H. Karlsen, Difference-quadrature schemes for non linear degenerate parabolic integro-PDE, SIAM J. Numer. Anal., 48 (2010), pp. 1110-1135

work page 2010

[8] [8]

Blessing, R

J. Blessing, R. Denk, M. Kupper, and M. Nendel. Convex monotone semigroups and their generators with respect to -convergence, J. Funct. Anal., 288 (8) (2025), 110841

work page 2025

[9] [9]

Blessing and M

J. Blessing and M. Kupper. Nonlinear semigroups built on generating families and their Lipschitz sets, Potential Analysis, 59 (2023), pp. 857-895

work page 2023

[10] [10]

Blessing and M

J. Blessing and M. Kupper, Nonlinear semigroups and limit theorems for convex expectations, Ann. Appl. Probab., 35(2) (2025), pp. 779-821

work page 2025

[11] [11]

Blessing, L

J. Blessing, L. Jiang, M. Kupper, and G. Liang, Convergence rates for Chernoff-type approximations of convex monotone semigroups, Stochastic Process. Appl., 189 (2025), 104700

work page 2025

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Chowdhury and E

I. Chowdhury and E. E. R. Jakobsen, Precise error bounds for numerical approximations of fractional HJB equations. IMA Journal of Numerical Analysis, 45(2), (2025), 633-672

work page 2025

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G. M. Coclite, O. Reichmann, and N. H. Risebro, A convergent difference scheme for a class of partial integro-differential equations modeling pricing under uncertainty, SIAM J. Numer. Anal., 54 (2016), pp. 588-605

work page 2016

[14] [14]

del Teso, J

F. del Teso, J. Endal, and E. R. Jakobsen, Robust numerical methods for nonlocal (and local) equations of porous medium type. Part I: Theory, SIAM J. Numer. Anal., 57 (2019), pp. 2266-2299

work page 2019

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R. Denk, M. Kupper, and M. Nendel, A semigroup approach to nonlinear L´ evy processes, Stochastic Process. Appl., 130 (2020), pp.1616-1642

work page 2020

[16] [16]

X. Fang, S. Peng, Q. Shao, and Y. Song, Limit theorem with rate of convergence under sublinear expectations, Bernoulli, 25 (2019), pp. 2564-2596

work page 2019

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M. Hu, L. Jiang, and G. Liang, On the rate of convergence for an α-stable central limit theorem under sublinear expectation, Journal of Applied Probability 62(1) (2025), pp. 209-233

work page 2025

[18] [18]

M. Hu, L. Jiang, G. Liang, and S. Peng, A universal robust limit theorem for nonlinear L´ evy processes under sublinear expectation, Probab. Uncertain. Quant. Risk, 8 (2023), pp. 1-32

work page 2023

[19] [19]

Hu and S

M. Hu and S. Peng, G-L´ evy processes under sublinear expectations, Probab. Uncertain. Quant. Risk, 6 (2021), pp. 1-22

work page 2021

[20] [20]

Huang and G

S. Huang and G. Liang, A monotone scheme for G-equations with application to the convergence rate of robust central limit theorem, Journal of Differential Equations, 398 (2024), pp. 1-37. 24

work page 2024

[21] [21]

I. A. Ibragimov and Y. V. Linnik, Independent and stationary sequences of random variables, Wolters- Noordhoff, Groningen, 1971

work page 1971

[22] [22]

E. R. Jakobsen, K. H. Karlsen, and C. La Chioma, Error estimates for approximate solutions to bellman equations associated with controlled jump-diffusions, Numer. Math., 110 (2008), pp. 221-255

work page 2008

[23] [23]

L. Jiang, Error estimates for the robust α-stable central limit theorem under sublinear expectation by a discrete approximation method, Journal of Mathematical Analysis and Applications 543(2) (2025), 129028

work page 2025

[24] [24]

Jiang and G

L. Jiang and G. Liang, A robust α-stable central limit theorem under sublinear expectation without integrability condition, Journal of Theoretical Probability, 37 (2024), pp. 2394-2424

work page 2024

[25] [25]

Kazi-Tani, D

N. Kazi-Tani, D. Possama¨ ı, and C. Zhou, Second-order BSDEs with jumps: formulation and uniqueness, Ann. Appl. Probab., 25 (2015), pp. 2867-2908,

work page 2015

[26] [26]

N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman’s equations, St. Petersburg Math. J., 9 (1997), pp. 639-650

work page 1997

[27] [27]

N. V. Krylov, Approximating value functions for controlled degenerate diffusion processes by using piece-wise constant policies, Electron. J. Probab., 4 (1999), pp. 1-19

work page 1999

[28] [28]

N. V. Krylov, On the rate of convergence of finite-difference approximations for Bellman’s equations with variable coefficients, Probab. Theory Related Fields, 117 (2000), pp. 1-16

work page 2000

[29] [29]

N. V. Krylov, On Shige Peng’s central limit theorem, Stochastic Process. Appl., 130 (2020), pp. 1426- 1434

work page 2020

[30] [30]

Kuhn, Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear L´ evy(- type) processes, ALEA, 16 (2019), pp

F. Kuhn, Viscosity solutions to Hamilton-Jacobi-Bellman equations associated with sublinear L´ evy(- type) processes, ALEA, 16 (2019), pp. 531-559

work page 2019

[31] [31]

Nendel and M

M. Nendel and M. R¨ ockner, Upper envelopes of families of Feller semigroups and viscosity solutions to a class of nonlinear Cauchy problems, SIAM Journal on Control and Optimization, 59 (2021), pp. 4400-4428

work page 2021

[32] [32]

Neufeld and M

A. Neufeld and M. Nutz, Nonlinear L´ evy processes and their characteristics, Trans. Amer. Math. Soc., 369 (2017), pp. 69-95

work page 2017

[33] [33]

Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itˆ o type, in: Stochastic Analysis and Applications, in: Abel Symp., vol

S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itˆ o type, in: Stochastic Analysis and Applications, in: Abel Symp., vol. 2, Springer, Berlin, 2007, pp. 541-567

work page 2007

[34] [34]

Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process

S. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 118(1) (2008), pp. 2223-2253

work page 2008

[35] [35]

Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probab

S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncer- tain. Quant. Risk, 4(1) (2019), pp. 1-8

work page 2019

[36] [36]

Peng, Nonlinear expectations and stochastic calculus under uncertainty, Springer, Berlin, 2019

S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, Springer, Berlin, 2019. 25

work page 2019

[37] [37]

Soner, N

H.M. Soner, N. Touzi, J. Zhang, Quasi-sure stochastic analysis through aggregation, Electron. J. Probab., 16 (2011), pp. 1844-1879

work page 2011

[38] [38]

Soner, N

H.M. Soner, N. Touzi, J. Zhang, Wellposedness of second order backward SDEs, Probab. Theory Related Fields, 153 (2012), pp. 149-190

work page 2012

[39] [39]

Soner, N

H.M. Soner, N. Touzi, J. Zhang, Dual formulation of second order target problems, Ann. Appl. Probab., 23 (2013), pp. 308-347

work page 2013

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Song, Normal approximation by Stein’s method under sublinear expectations, Stochastic Process

Y. Song, Normal approximation by Stein’s method under sublinear expectations, Stochastic Process. Appl., 130 (2020), pp. 2838-2850. 26

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