Stability Analysis of 4-Stage Stochastic Runge-Kutta Method (SRK4) and Specific Stochastic Runge-Kutta Method (SRKS1.5) for Stochastic Differential Equations

  • Noor Amalina Nisa AriffinEmail author
  • Norhayati Rosli
  • Abdul Rahman Mohd Kasim
Conference paper


This paper is devoted to investigate the mean-square stability of 4-stage stochastic Runge-Kutta (SRK4) and specific stochastic Runge-Kutta (SRKS1.5) methods for linear stochastic differential equations (SDEs). The mean-square stability functions of SRK4 and SRKS1.5 are derived. The regions in which the methods are stable in the mean-square sense are plotted. Numerical experiments are performed to verify the stability properties of both methods.


Stochastic differential equations Stochastic Runge-kutta Mean square stable Explicit method General mean square stable 



We would like to thank the Ministry of Education (MOE) and Research and Innovation Department, Universiti Malaysia Pahang (UMP) for their financial supports through FRGS Vote No: RDU130122 and Internal UMP Grant RDU1703190.


  1. 1.
    Norhayati, R., Arifah, B., Yeak, S.H., Haliza, A.R., Madihah, M.S.: Performance of Euler-Maruyama, 2-stage SRK and 4- stage SRK in approximating the strong solution of stochastic model. Sains Malaysiana 39, 851–857 (2010)Google Scholar
  2. 2.
    Xiao, A., Tang, X.: High strong order stochastic Runge-kutta method for stratonovich stochastic differential equations with scalar noise. Springer, New York (2015)zbMATHGoogle Scholar
  3. 3.
    Milstein, G.N.: Numerical Integration of Stochastic Differential Equations. Springer, Berlin (1995)CrossRefGoogle Scholar
  4. 4.
    Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)CrossRefGoogle Scholar
  5. 5.
    Burrage, K., Burrage, P.M.: High strong order explicit Runge-Kutta methods for stochastic ordinary differential equations. Appl. Numer. Math. 22(1–3), 81–101 (1996)Google Scholar
  6. 6.
    Saito, Y., Mitsui, T.: Stability analysis of numerical schemes for stochastic differential equations. SIAM J. Num. Anal. 33(6), 2254–2267 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Burrage, P.M.: Runge-Kutta methods for stochastic differential equations. Ph.D. thesis, University of Queensland, Australia (1999)Google Scholar
  8. 8.
    Higham, D.J.: Mean-square and asymptotic stability of the stochastic theta method. SIAM J. Numer. Anal. 38(3), 753–769 (2000)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Saito, Y., Mitsui, T.: Mean-square stability of numerical schemes for stochastic differential systems. Vietnam J. Math. 30, 551–560 (2002)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Liu, M.Z., Spijker, M.N.: The stability of the θ-methods in the numerical solution of delay differential equations. IMA J. Numer. Anal. 10(1), 31–48 (1990)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ryashko, L.B., Schurz, H.: Mean square stability analysis of some linear stochastic systems. Dyn. Syst. Appl. 6, 165–190 (1997)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  • Noor Amalina Nisa Ariffin
    • 1
    Email author
  • Norhayati Rosli
    • 1
  • Abdul Rahman Mohd Kasim
    • 1
  1. 1.Faculty of Industrial Sciences & TechnologyUniversiti Malaysia PahangGambangMalaysia

Personalised recommendations