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

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

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

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