Stokastik Diferansiyel Denklemlerin Ontolojisi

Abstract

Bu çalışma, stokastik diferansiyel denklemlerin (SDE’ler) matematiksel formülasyonları, ontolojik temelleri ve uygulama alanlarının kapsamlı bir incelemesini sunmaktadır. SDE’ler, belirsizlik ve rastgelelik gibi karmaşık olguların modellenmesinde kritik bir rol oynar ve finansal piyasalardan biyolojik sistemlere kadar çok çeşitli alanlarda uygulanabilir. Makale, Itô ve Stratonovich hesabının matematiksel yaklaşımlarını karşılaştırarak SDE’lerin çözüm yöntemlerini ve teorik temellerini ayrıntılı olarak açıklamaktadır. Ek olarak, SDE’lerin ontolojik temelleri ve çeşitli bilimsel ve mühendislik alanlarındaki uygulamaları incelenmektedir. Finans, biyoloji, kriptoloji ve blok zinciri teknolojisindeki kullanımlarına özel vurgu yapılmaktadır. Sonuçlar, SDE’lerin matematiksel modellemedeki önemini ve çok sayıda uygulama alanındaki etkilerini vurgulamaktadır.

Keywords

• Stokastik Diferansiyel Denklemler
• Itô İntegrali
• Stratonovich İntegrali
• Matematiksel Modelleme.

Ontology of Stochastic Differential Equations

Abstract

This study provides a comprehensive examination of the mathematical formulations, ontological foundations, and application domains of stochastic differential equations (SDEs). SDEs play a critical role in modeling complex phenomena such as uncertainty and randomness and can be applied across a wide range of fields from financial markets to biological systems. The paper contrasts the mathematical approaches of Itô and Stratonovich calculus, detailing the solution methods and theoretical foundations of SDEs. Additionally, the ontological foundations of SDEs and their applications in various scientific and engineering fields are explored. Emphasis is placed on their use in finance, biology, cryptology, and blockchain technology. The results highlight the significance of SDEs in mathematical modeling and their impact across numerous application areas.

Keywords

• Stochastic Differential Equations
• Itô Integral
• Stratonovich Integral
• Mathematical Modeling.

References

1. Ito K. On Stochastic Differential Equations. Mem Amer Math Soc, 1951; 4: 1-51.
2. Stratonovich RL. A New Representation for Stochastic Integrals and Equations. SIAM Journal on Control, 1966; 4: 362-371.
3. Karatzas SE, Shreve SE. Brownian Motion and Stochastic Calculus. 2nd Ed. USA: Springer-Verlag; 1991.
4. Allen LJS. An Introduction to Stochastic Processes with Applications to Biology, USA: Springer; 2008.
5. Itô K, McKean HP. Diffusion Processes and their Sample Paths, USA: Springer; 1965.
6. Rivest RL, Adleman L, Deaouzos ML. On Data Banks and Privacy Homomorphism. In: DeMillo, RA, editors. Foundations of Secure Computation, USA: Academic Press, 1978: 169-179.
7. Nakamoto S. Bitcoin: A Peer-to-Peer Electronic Cash System; 2008. https://bitcoin.org/bitcoin.pdf
8. Kang H, Chang X, Mišić J, Mišić VB, Yao Y, Chen Z. Stochastic modeling approaches for analyzing blockchain: A survey. arXiv preprint arXiv:2009.05945, 2020.

9. Black F, Scholes M. The pricing of options and corporate liabilities. Journal of Political Economy, 1973; 81(3): 637-654.
10. Frigg R, Hartmann S. Models and Fiction. In The Philosophy of Model-Based Science, USA: Springer, 2012; 49-97.
11. Gardiner CW. Handbook of Stochastic Methods, USA: Springer, 2004.
12. Hull JC. Options, Futures, and Other Derivatives, AUSTRALIA: Pearson 2015.
13. Van Kampen NG. Stochastic differential equations, Physics Reports, 1976; 24(3): 171-228.
14. Merton RC. Theory of rational option pricing. Bell J Econ Manage Sci, 1973; 4(1): 141-183.
15. Nelkin D. Stochastic Processes and Applications. USA: Academic Press, 1983.
16. Norton JD. Mathematical Models in the Sciences: The Role of Simulation anda Computation. MIT Press, 2019.
17. Øksendal B. Stochastic Differential Equations, USA: Springer-Verlag, 2000.
18. Kuznetsov DF. Mean-Square Approximation of Iterated Itˆo and Stratonovich Stochastic Integrals: Method of Generalized Multiple Fourier Series. Application to Numerical Integration of Itˆo SDEs and Semilinear SPDEs. Differential Equations and Control Processes, 2021; 4: 4-14.