Regularity of Generalized Mean-Field G-SDEs
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsSee the attached file.
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Author Response
Thank you for your comments and suggestions. Please, find our response attached.
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Reviewer 2 Report
Comments and Suggestions for AuthorsPlease revise as per the comments provided in the attachment.
Comments for author File: Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 3 Report
Comments and Suggestions for AuthorsThe manuscript investigates the regularity properties of solutions to generalised mean-field stochastic differential equations (SDEs) under the G-expectation framework. Specifically, the authors establish the first and second order Fr\'echet differentiability of the solution with respect to the stochastic initial condition, and they further characterise the corresponding Fr\'echet derivatives via associated G-SDEs. The paper builds upon recent developments in mean-field SDEs and the G-framework introduced by Peng, and extends existing results on differentiability to this non-linear and sublinear expectation setting. The results are mathematically rigorous and technically well-developed.
The paper is well-written, logically structured, and makes a meaningful contribution to the literature on stochastic analysis under uncertainty. The novelty lies in the systematic study of regularity for mean-field G-SDEs, which is of significant interest for both theoretical development and applications in finance, physics, and applied probability. The manuscript is technically solid and the mathematical arguments appear correct. I believe the paper is suitable for publication after minor revisions.
- Please expand the abstract by adding more detail on the main contributions. For example, explicitly mention the specific results on first and second order Fr\'echet differentiability and their potential applications.
- The introduction provides a strong background on mean-field SDEs and the G-framework. However, I suggest adding a brief discussion on how the derived regularity results could be concretely applied in stochastic control, mean-field games, or numerical methods, to highlight the practical relevance.
- Several references in the introduction are cited only by numbers without explanation (e.g., [24], [25], [27]). It would be helpful to briefly summarise their contributions.
- There are a few minor typographical inconsistencies, for instance in the equation numbering and spacing around displayed formulas. Please check carefully.
- Consider expanding the conclusion to emphasise the significance of the results and outline potential directions for future research.
Author Response
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Reviewer 4 Report
Comments and Suggestions for AuthorsSee attachement
Comments for author File: Comments.pdf
Author Response
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Author Response File: Author Response.pdf
Reviewer 5 Report
Comments and Suggestions for AuthorsReview on the article “Regularity of Generalised Mean-Field G-SDEs» by Karl-Wilhelm Georg Bollweg and Thilo Meyer-Brandis
The authors of the reviewed article investigated the regularity properties of the solution of a stochastic differential equation governed by G-Brownian motion. The Fréchet differentiability of the solution with respect to a random initial condition was studied. The Fréchet derivatives of the first and second order were considered. Perturbations of the initial data propagate through the stochastic system. The sensitivity of the solution with respect to changes in the initial data is studied, which is an important aspect in a wide range of applications. The authors developed the concept of differentiability for mappings on the space of sublinear distributions. To obtain the main results, tested approaches (approximation theory, Grönwall inequalities, etc.) and combinations of such approaches to a new type of problems were applied.
The results of the work look new, sufficiently substantiated.
Some comments on the text and wishes to the authors.
- Insert Keywords, MSC 2000.
- In the abstract, briefly (1-2 sentences) dwell on the methods of obtaining the results announced in the article.
- Of course, the structure of the article is the right of the authors. But I lack a final part at the end (for example, in the Conclusions), in which the authors could clearly indicate the place and theoretical significance of the obtained results in the general theory of SDEs, possible generalizations and limitations.This is just a wish.
In general, the article makes a rather positive impression. I suggest the authors improve the text and presentability of the obtained results.
Author Response
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