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

Yıl 2020, Cilt: 13 Sayı: 2, 898 - 916, 31.08.2020

Hande Akdemir , Dudu Aydın Oğur

https://doi.org/10.18185/erzifbed.617161

Öz

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.

Anahtar Kelimeler

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

Destekleyen Kurum

Giresun Üniversitesi Bilimsel Araştırma Projeleri Birimi

Proje Numarası

FEN-BAP-A-230218-49

Teşekkür

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

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

Öz

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

Anahtar Kelimeler

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

Proje Numarası

FEN-BAP-A-230218-49

Kaynakça

• 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.
• 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.
• 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.
• 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.
• Fehlberg, E. (1969). “Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems”, NASA Technical Report 315.
• 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.
• Ito, K. (1951). “On Stochastic Differential Equations”, Memoirs of the American Mathematical Society, 4, 1-51.
• 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.
• 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.
• Komori, Y., Buckwar, E. (2013). “Stochastic Runge-Kutta methods with deterministic high order for ordinary differential equations”, BIT Numerical Mathematics, 53(3), 617-639.
• 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.
• Langevin, P. (1908). “Sur la théorie du mouvement brownien”, Comptes Rendus, 146, 530-533.
• Maruyama, G. (1955). “Continuous Markov processes and stochastic equations”, Rendiconti del Circolo Matematico di Palermo, 4(1), 48-90.
• Milstein, G. N. (1994). “Numerical integration of stochastic differential equations”, Vol. 313, Springer Science and Business Media.
• 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.
• 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.
• 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.
• 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.
• 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.
• Rößler, A. (2009). “Second order Runge–Kutta methods for Itô stochastic differential equations”, SIAM Journal on Numerical Analysis, 47(3), 1713-1738.
• Rümelin, W. (1982). “Numerical treatment of stochastic differential equations”, SIAM Journal on Numerical Analysis, 19(3), 604-613.
• 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.
• 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.
• 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.
• Wang, P. (2015). “A-stable Runge–Kutta methods for stiff stochastic differential equations with multiplicative noise”, Computational and Applied Mathematics, 34(2), 773-792.
• Wiener, N. (1923). “Differential‐Space”, Journal of Mathematics and Physics, 2(1-4), 131-174.