Sains Malaysiana 39(5)(2010): 851–857

Performance of Euler-Maruyama, 2-Stage SRK and 4-Stage SRK in Approximating the Strong Solution of Stochastic Model

(Keberkesanan Kaedah Euler-Maruyama, Stokastik Runge-Kutta Peringkat 2, Stokastik

Runge-Kutta Peringkat 4 dalam Mencari Penyelesaian Penghampiran Model Stokastik)

Norhayati Rosli*, Arifah Bahar, Yeak Su Hoe & Haliza Abdul Rahman

Department of Mathematics, Faculty of Science

Universiti Teknologi Malaysia, 81310 UTM Skudai, Johor, Malaysia

Madihah Md. Salleh

Faculty of Bioscience and Bioengineering, Universiti Teknologi Malaysia

81310 UTM Skudai, Johor, Malaysia

Received: 13 August 2009 / Accepted: 19 March 2010

ABSTRACT

Stochastic differential equations play a prominent role in many application areas including finance, biology and epidemiology. By incorporating random elements to ordinary differential equation system, a system of stochastic differential equations (SDEs) arises. This leads to a more complex insight of the physical phenomena than their deterministic counterpart. However, most of the SDEs do not have an analytical solution where numerical method is the best way to resolve this problem. Recently, much work had been done in applying numerical methods for solving SDEs. A very general class of Stochastic Runge-Kutta, (SRK) had been studied and 2-stage SRK with order convergence of 1.0 and 4-stage SRK with order convergence of 1.5 were discussed. In this study, we compared the performance of Euler-Maruyama, 2-stage SRK and 4-stage SRK in approximating the strong solutions of stochastic logistic model which describe the cell growth of C. acetobutylicum P262. The MS-stability functions of these schemes were calculated and regions of MS-stability are given. We also perform the comparison for the performance of these methods based on their global errors.

Keywords: 2-stage stochastic Runge-Kutta; 4-stage stochastic Runge-Kutta; Euler-Maruyama; stochastic differential equations

ABSTRAK

Persamaan pembezaan stokastik memainkan peranan penting dalam kebanyakan bidang seperti kewangan, biologi dan epidemiologi. Dengan menggabungkan elemen rawak terhadap sistem persamaan pembezaan biasa, pesamaan pembezaan stokastik muncul. Ini membawa kepada fenomena fizikal yang lebih kompleks berbanding dengan persamaan deteministik yang setara dengannya. Walau bagaimanapun, persamaan pembezaan stokastik tidak mempunyai penyelesaian analitik dan kaedah penyelesaian berangka merupakan cara terbaik untuk mengatasi masalah ini. Pada abad ini, banyak usaha telah dilakukan untuk mencari penyelesaian hampiran persamaan pembezaan stokastik. Bentuk am kelas Stokastik Runge-Kutta, SRK telah dikaji dan secara khusunya SRK peringkat 2 dengan pangkat penumpuan 1.5 dan SRK peringkat 4 dengan pangkat penumpuan 2.0 telah dibincangkan. Dalam kajian ini, kami melakukan perbandingan bagi melihat keberkesanan kaedah Euler-Maruyama, SRK peringkat 2 dan SRK peringkat 4 bagi mencari penyelesaian hampiran ke atas model logistik stokastik yang menerangkan kadar pertumbuhan sel C. acetobutylicum P262. Keberkesanan kaedah tersebut telah dibandingkan berdasarkan analisis stabiliti min kuasa dua dan ralat sejagat.

Kata kunci: Euler-Maruyama; persamaan pembezaan stokastik; stokastik Runge-Kutta peringkat 2; stokastik Runge-Kutta peringkat 4

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*Corresponding author; e-mail: norhayati@ump.edu.my