Analysis of Noise on Ordinary and Fractional-Order Financial Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors

Please see our comments

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear all,

The article analyzes the influence of stochastic fluctuations on financial system stability, comparing the behavior of ordinary and fractional-order financial models in the presence of noise. The authors use numerical simulations in MATLAB to compare system responses in deterministic and noisy environments, highlighting fundamental differences between integer-order and fractional derivatives in financial modeling. The main contributions consist of demonstrating the increased sensitivity of fractional systems to external shocks and emphasizing the importance of memory effects in financial modeling for improving accuracy in predicting market behavior.

General Comments

The article addresses a relevant and current topic in financial modeling, exploring the impact of stochastic noise on financial systems. The comparative perspective between ordinary and fractional-order systems provides a valuable contribution to understanding the behavior of financial markets under uncertainty.

From a conceptual standpoint, the paper presents a solid approach, building upon existing models and extending them to include noise and memory effects. The theoretical framework is well-founded, with clear explanations of Caputo fractional derivatives and fractional Gaussian noise (FGN).

The work is particularly timely given the increasing volatility in global financial markets and growing interest in advanced mathematical tools for risk assessment.

Several limitations that require attention before publication:

• Empirical validation: Although the numerical simulations are rigorously performed, empirical validation using real financial data is missing. The authors mention this limitation in the future directions section, but should clarify this aspect more explicitly as a limitation of the current study. This could be addressed by at least testing the model against historical market data from a recent financial crisis event.
• Parameter justification: The choice of system parameters (a, b, c, d, g) and noise intensity is taken from existing literature, but a sensitivity analysis would be beneficial to demonstrate the robustness of results to variations in these parameters. I recommend adding a subsection in Section 3 dedicated to sensitivity analysis, exploring how slight variations in key parameters affect overall system behavior.
• Noise comparability: While the authors compare an ordinary system with normal noise to a fractional system with FGN, it would be useful to discuss the effects of normal noise on the fractional system for a more complete comparison. This additional comparison would strengthen the methodological approach and provide a more comprehensive understanding of noise effects.
• Practical applicability: The financial interpretation section offers valuable perspectives, but could be better connected with concrete financial situations and recent developments in global financial markets. Specifically, connecting the model findings to events like the 2020 pandemic market shock or cryptocurrency volatility would enhance relevance.

Specific Comments

• Lines 44-48: The introduction of fractional derivatives is rigorous, but a more intuitive explanation of why they are suitable for modeling financial systems would be useful.
• Lines 54-57: The explanation regarding fractional Gaussian noise (FGN) is technical but could benefit from clarification of the financial interpretation of the Hurst exponent for readers less familiar with these concepts.
• Figures 1-4: The graphs would benefit from clearer legends and more descriptive axis labels. It would also be useful to include a direct comparison between ordinary and fractional system behavior on the same graph for key variables.
• Lines 162-165: The claim regarding the "more structured" behavior of the fractional system requires a more detailed explanation or additional references for support.
• Lines 187-189: The conclusion regarding the trade-off between simplicity and realism is valuable but could be elaborated with concrete examples from practice.
• Lines 219-224: The implications for monetary policy could be developed with references to current instruments and practices of central banks.
• References section: The bibliography could be updated with more recent works (from the last 5 years) in the field of behavioral finance and complex systems.

 Good luck!

 

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors Why were bounded noise and fractional Gaussian noise used at all? Could you explain the reason a little more?
The numerical method used in the simulations is explained too briefly. Shouldn't more detail be provided about it?
The difference in behavior between conventional and fractional systems is not explained well in the text. Shouldn't they explore this part a little deeper?
The importance of long-term memory in fractional models is not addressed very well. Shouldn't they highlight this issue more?
Are the suggestions for future research, such as using mixed models, really practical? Are more details provided or are they left too superficial?
The impact of noise on important variables such as interest rates and profit margins is not fully explored. Shouldn't they analyze this part a little more?
The role of monetary policy and risk management in reducing financial instability is not explained too well. Shouldn't they explain this more?
The comparison between conventional and fractional models is not very clear. Couldn't it be made simpler and more understandable?
Are only numerical simulations used or are real data also used for validation? If not, why not?   Comments on the Quality of English Language

The language of the article needs to be revised.

Author Response

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Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

In the context of modern financial markets characterized by high volatility, mathematical modeling and quantitative analysis have become essential tools for forecasting and developing effective strategies in portfolio management. By applying advanced mathematical methods, such as stochastic processes, optimization algorithms, and risk assessment models like Value at Risk, financial professionals can enhance decision-making processes, optimize asset allocations, and mitigate potential risks associated with investment portfolios.

In this regard, the peer-reviewed study examining the impact of stochastic fluctuations on the stability of financial systems is relevant. The paper explores this influence through both ordinary and fractional-order financial models under the presence of noise, making it a significant contribution to the field.

Below, I present a brief review of the work along with my comments.

In the introduction, the authors provide a brief overview of the Caputo fractional derivative and Gaussian noise, focusing on their applications in the analysis of financial systems and the models discussed in the subsequent sections of the paper. This overview is supported by appropriate references to the relevant literature.

The model equations are solved numerically in Section 3 for equations (9) with the ordinary time derivative of the first order and in Section 4 for the Caputo time derivative in equations (11) both in the absence and presence of noise in order to identify the influence of noise on the evolution of the model variables.

An analysis of the obtained numerical solutions is given in Section 5. It is noted that due to the non-local memory effect due to the Caputo fractional derivative the system is more responsive to persistent stochastic inputs. The need to take into account noise in the model equations is obvious.

Based on the obtained numerical solutions, the authors provide a comparative description of the characteristic features of the dynamic modes in the models under consideration. In particular, it is argued that the ordinary system shows resilience and rapid return to equilibrium, while the fractional one is more sensitive and requires longer to stabilize. Fractional-order models capture long-term dependencies in financial behavior, while ordinary models may suit short-term forecasting where historical influence is limited.

Sections 6–8 discuss the application of the results to model financial systems and further prospects for generalization and extension of financial applications.

Comments (optional).

1. The authors should apply the model they propose to real (already published) data on trading of financial instruments on the stock exchange, to the dynamics of stock indices. Finally, it would be of particular interest to apply the results of the work to stock quotes of digital currency.
2. It would also be interesting to adapt the models used by the authors so that it would be possible to compare the results of these models with the well-known Black and Scholes type models for financial instruments in the context of fractional analysis (reference [28] in the list of references).

Overall, the reviewed work makes a certain useful contribution to the development of mathematical methods in financial analysis. I believe that the reviewed work can be recommended for acceptance in the Fractal and Fractional journal, taking into account the comments made.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors

You did a good job.

Thanks

Reviewer 3 Report

Comments and Suggestions for Authors

I have no other comment.