Analytical and Computational Analysis of Fractional Stochastic Models Using Iterated Itô Integrals
Round 1
Reviewer 1 Report
This is a comprehensive study of two applications of the Fractional Brownian Motion (fBM), respectively in analysing the Black-Scholes model (analytical study) and a population biology model (numerical study), that, in the limit considered, equates to a Logistic model. In their analysis, the authors have first structured their model in line with the standard Rieman-Liouville (RL) model, followed by the Grunwald-Letnikov (GL) to account for the limitations of the RL model. Their results conform to the known results in the asymptotic limits. The article has been mostly written in clear and precise English, barring occasional digressions (can be corrected through self-study).
I consider this a valuable contribution in a wide range of fields that involve dynamical systems and reaction-diffusion models. Before accepting, I would suggest that the authors consider the following improvements options:
1. Since the title refers to Ito calculus, there should be an explanation on why this method is selectively applied to Ito models. For example, what about Stratonovich calculus?
2. The authors mention that the models can address memory effects. This needs to be clarified as past studies indicate that inclusion of memory effects typically ends with integral equations that can only be numerically solved. If so, what is the advantage?
3. I would encourage a discussion on a minor extension of the Black-Scholes model for the case of Ornstein-Uhlenbeck, i.e. time-dependent noise. Would you still arrive at such closed form solutions? This is relevant as past studies, e.g. Prob Eng Mech 15, 131 (2000); PRE 94, 022139 etc have studied such effects and have arrived at answers using alternative techniques.
4. The literature survey needs to include comparisons with past studies, e.g. the two articles mentioned above (but not limited to only those two).
5. I would recommend moving a substantial section of the theoretical components from Section 2 to an Appendix as many of the results shown are textbook deliberations.
This is a good work that can be improved further by addressing the points detailed above.
Perfectly acceptable English; minor digressions noted that can be improved through self-study. No major editing is required.
Author Response
Many thanks for reading the manuscript and for suggestions that help to improve the quality and readability. We have considered all the comments as shown below in details.
- Since the title refers to Ito calculus, there should be an explanation on why this method is selectively applied to Ito models. For example, what about Stratonovich calculus?
Done: The study is designed to be consistent with Ito model to be able for analysis with a unified sense. Analysis with Stratonovich calculus is also valid after some considerations or simply by re-writing models in Ito sense. Conversion from Stratonovich to Ito sense is well-known. A statement is added, after equation (17), to clarify this issue.
- The authors mention that the models can address memory effects. This needs to be clarified as past studies indicate that inclusion of memory effects typically ends with integral equations that can only be numerically solved. If so, what is the advantage?
Reply: Memory effect is included implicitly by considering the fractional derivatives/integrals. Our analysis also shows that we have a set of integral equations that can be solved analytically for some models and numerically in general. So, the current analysis is consistent with memory inclusion but with additional stochastic effects.
- I would encourage a discussion on a minor extension of the Black-Scholes model for the case of Ornstein-Uhlenbeck, i.e. time-dependent noise. Would you still arrive at such closed form solutions? This is relevant as past studies, e.g. Prob Eng Mech 15, 131 (2000); PRE 94, 022139 etc have studied such effects and have arrived at answers using alternative techniques.
Done: A paragraph is added before Model 2 to describe the FWHE solution of the Ornstein-Uhlenbeck process.
- The literature survey needs to include comparisons with past studies, e.g. the two articles mentioned above (but not limited to only those two).
Done: A paragraph is added to the introduction to refer the reader for other techniques in the literature. The reference PRE 94, 022139 is added to the list of references.
- I would recommend moving a substantial section of the theoretical components from Section 2 to an Appendix as many of the results shown are textbook deliberations.
Done: We have already let the reader refer to some references for the details. But we keep only the introductory material required for our analysis. We have to introduce both the for fractional calculus in addition to the stochastic processes. However, we have reduced the introduction.
This is a good work that can be improved further by addressing the points detailed above.
Many thanks for the valuable comments.
Regards
Authors, June 2023.
Reviewer 2 Report
The Abstract has not been written adequately and should be modified significantly. Indeed, the Abstract should reflect the overall content of the paper. Hence, the authors should rewrite the Abstract, not to detail it, but to focus on the outlines and the tasks in the paper without going into details.
Mathematical Background is too long and tedious.
There are two 3.1.
There are two 4.1.
The conclusions should be 5, not 4.
Please provide the limitations and advantages of your proposed scheme.
In my opinion, the authors provide some comparison results between the established method in the paper and other published approach for clear understanding of the main contribution.
The conclusion can also add some further research content to enhance the continuity of the research in this aspect.
The paper should be polished from an English writing point of view. Also, the equations and the results should be checked.
The paper should be polished from an English writing point of view.
Author Response
Many thanks for reading the manuscript and for suggestions that help to improve the quality and readability. We have considered all the comments as shown below in details.
- The Abstract has not been written adequately and should be modified significantly. Indeed, the Abstract should reflect the overall content of the paper. Hence, the authors should rewrite the Abstract, not to detail it, but to focus on the outlines and the tasks in the paper without going into details.
Done: The Abstract is re-written to reflect the paper content.
- Mathematical Background is too long and tedious.
Done: The introduction is reduced as recommended. We have already let the reader refer to some references for the details. But we keep only the introductory material required for our analysis. We have to introduce both the for fractional calculus in addition to the stochastic processes.
- There are two 3.1. There are two 4.1. The conclusions should be 5, not 4.
Done: The enumerations are reviewed and corrected.
- Please provide the limitations and advantages of your proposed scheme.
Done: The paragraph, before conclusions, is modified to assure the advantages and drawbacks in using proposed scheme.
- In my opinion, the authors provide some comparison results between the established method in the paper and other published approach for clear understanding of the main contribution.
- The conclusion can also add some further research content to enhance the continuity of the research in this aspect.
Done: A new paragraph is added to the conclusions for the future extensions of the current work.
- The paper should be polished from an English writing point of view. Also, the equations and the results should be checked.
Done: The English is revised again. Also, the equations and results are checked again.
Many thanks for the valuable comments.
Regards
Authors, June 2023
Reviewer 3 Report
See attch
Comments for author File: 
Comments.pdf
Author Response
Dear Respected Reviewer
Many thanks for the helpful comments that help to enhance the quality of the manuscript. All the comments are replied as required.
Regards
Authors
Author Response File: 
Author Response.docx
