A Stochastic Runge-Kutta-Fehlberg Method for Itô Stochastic Differential Equations with Small Noise

Abstract

In this study, a stochastic Runge-Kutta-Fehlberg (SRKF) method is proposed for the Itô stochastic differential equations (SDE) with a small factor in the diffusion coefficient. This method, which uses mixed stochastic (classical-stochastic) integrals, is an extension of derivative-free six-stage RKF method which is well known for deterministic DE. In intermediate steps of the proposed method, the Euler-Maruyama predictor is used. For some test problems, simulation studies are conducted to examine strong convergence of the method and compare it with some known methods

Keywords

• Runge-Kutta-Fehlberg method,
• Itô stochastic differential equations
• numerical solutions

Project Number

FEN-BAP-A-230218-49

Küçük Gürültü Terimi İçeren Itô Stokastik Diferansiyel Denklemler için Stokastik Runge-Kutta-Fehlberg Yöntemi

Abstract

Bu çalışmada, difüzyon teriminde küçük bir çarpan olan Itô stokastik diferansiyel denklemler (SDD) için stokastik Runge-Kutta-Fehlberg yöntemi (SRKFY) önerilmiştir. Bu yöntem, deterministik diferansiyel denklemler için iyi bilinen ve türevleri kullanmayan altı aşamalı RKFY’nin karışık stokastik (klasik-stokastik) integralleri kullanan bir uyarlamasıdır. Önerilen yöntemin ara adımlarında Euler-Maruyama tahminleyicisi kullanılmıştır. Bazı test problemleri için, yöntemin kuadratik orta anlamda yakınsaklığını incelemek ve bilinen bazı yöntemlerle karşılaştırmak amacıyla simülasyon çalışmaları yapılmıştır.

Keywords

• Runge-Kutta-Fehlberg yöntemi
• Itô stokastik diferansiyel denklemler
• nümerik çözümler
• küçük gürültü

Supporting Institution

Giresun Üniversitesi Bilimsel Araştırma Projeleri Birimi

Project Number

FEN-BAP-A-230218-49

Thanks

Bu çalışma, Giresun Üniversitesi Bilimsel Araştırma Projeleri Birimi tarafından desteklenmiştir (Proje No: FEN-BAP-A-230218-49).

References

1. Averina, T. A., Artemiev, S. S., and Schurz, H. (1994). “Simulation of stochastic auto-oscillating systems through variable stepsize algorithms with small noise”, Weierstraß-Institut für Angewandte Analysis und Stochastik, Berlin, Preprint 116.
2. Brown, R. (1828). “On the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies”, Edinburgh New Philosophical Journal, 5, 358-371.
3. Buckwar, E., Rößler, A., and Winkler, R. (2010). “Stochastic Runge–Kutta methods for Itô SODEs with small noise”, SIAM Journal on Scientific Computing, 32(4), 1789-1808.
4. Einstein, A. (1905). “Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen”, Annalen der physik, 322(8), 549-560.
5. Fehlberg, E. (1969). “Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems”, NASA Technical Report 315.
6. Hu, G., Wang, K. (2011). “The estimation of probability distribution of SDE by only one sample trajectory”, Computers and Mathematics with Applications, 62(4), 1798-1806.
7. Ito, K. (1951). “On Stochastic Differential Equations”, Memoirs of the American Mathematical Society, 4, 1-51.
8. Komori, Y. (2007). “Weak second-order stochastic Runge–Kutta methods for non-commutative stochastic differential equations”, Journal of Computational and Applied Mathematics, 206(1), 158-173.

9. Komori, Y. (2008). “Weak first or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process”, Journal of Computational and Applied Mathematics, 217(1), 166-179.
10. Komori, Y., Buckwar, E. (2013). “Stochastic Runge-Kutta methods with deterministic high order for ordinary differential equations”, BIT Numerical Mathematics, 53(3), 617-639.
11. Komori, Y., Cohen, D., and Burrage, K. (2017). “Weak Second Order Explicit Exponential Runge--Kutta Methods for Stochastic Differential Equations”, SIAM Journal on Scientific Computing, 39(6), A2857-A2878.
12. Langevin, P. (1908). “Sur la théorie du mouvement brownien”, Comptes Rendus, 146, 530-533.
13. Maruyama, G. (1955). “Continuous Markov processes and stochastic equations”, Rendiconti del Circolo Matematico di Palermo, 4(1), 48-90.
14. Milstein, G. N. (1994). “Numerical integration of stochastic differential equations”, Vol. 313, Springer Science and Business Media.
15. Milstein, G. N., Tretyakov, M. V. (1997a). “Mean-square numerical methods for stochastic differential equations with small noises”, SIAM Journal on Scientific Computing, 18(4), 1067-1087.
16. Milstein, G. N., Tretyakov, M. V. (1997b). “Numerical methods in the weak sense for stochastic differential equations with small noise”, SIAM journal on numerical analysis, 34(6), 2142-2167.
17. Milstein, G., Tretyakov, M. (2000). “Numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations”, Mathematics of Computation, 69(229), 237-267.
18. Newton, N. J. (1991). “Asymptotically efficient Runge-Kutta methods for a class of Ito and Stratonovich equations”, SIAM Journal on Applied Mathematics, 51(2), 542-567.
19. Rößler, A. (2005, December). “Explicit order 1.5 schemes for the strong approximation of Itô stochastic differential equations”, In PAMM: Proceedings in Applied Mathematics and Mechanics (Vol. 5, No. 1, pp. 817-818), Berlin: WILEY‐VCH Verlag.
20. Rößler, A. (2009). “Second order Runge–Kutta methods for Itô stochastic differential equations”, SIAM Journal on Numerical Analysis, 47(3), 1713-1738.
21. Rümelin, W. (1982). “Numerical treatment of stochastic differential equations”, SIAM Journal on Numerical Analysis, 19(3), 604-613.
22. Sickenberger, T., Weinmüller, E., and Winkler, R. (2009). “Local error estimates for moderately smooth problems: Part II—SDEs and SDAEs with small noise”, BIT Numerical Mathematics, 49(1), 217-245.
23. Valinejad, A., Hosseini, S. M. (2010). “A variable step-size control algorithm for the weak approximation of stochastic differential equations”, Numerical Algorithms, 55(4), 429-446.
24. Valinejad, A., Hosseini, S. M. (2012). “A stepsize control algorithm for SDEs with small noise based on stochastic Runge–Kutta Maruyama methods”, Numerical Algorithms, 61(3), 479-498.
25. Wang, P. (2015). “A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise”, Computational and Applied Mathematics, 34(2), 773-792.
26. Wiener, N. (1923). “Differential‐Space”, Journal of Mathematics and Physics, 2(1-4), 131-174.