A Study on Stability for Stochastic Differential Equations

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DOI: https://doi.org/10.28924/2291-8639-24-2026-67
Published: Mar 12, 2026
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Jarunee Sirisin, Kulandhaivel Karthikeyan, Sivaranjani Ramasamy, Sadhasivam Poornima, Panjaiyan Karthikeyann, Thanin Sitthiwirattham, A Study on Stability for Stochastic Differential Equations, Int. J. Anal. Appl., 24 (2026), 67.

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Jarunee Sirisin, Kulandhaivel Karthikeyan, Sivaranjani Ramasamy, Sadhasivam Poornima, Panjaiyan Karthikeyann, Thanin Sitthiwirattham

Abstract

This paper contributes a non-linear stability analysis for a class of stochastic Runge-Kutta algorithms by developing mean-square contractive solutions. This paper illustrates how the stochastic perturbation of a (k, l)- algebraically stable deterministic Runge-Kutta technique takes over this method and the solutions obtained by it. The numerical examples back up the validity of the conclusions.

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References

  1. K. Sobczyk, Stochastic Differential Equations: With Applications to Physics and Engineering, Springer, Dordrecht, 1991. https://doi.org/10.1007/978-94-011-3712-6.
  2. P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, Heidelberg, 1999. https://doi.org/10.1007/978-3-662-12616-5.
  3. X. Mao, Stochastic Differential Equations and Applications, Woodhead Publishing, 2008.
  4. G.G. Dahlquist, A Special Stability Problem for Linear Multistep Methods, BIT Numer. Math. 3 (1963), 27–43. https://doi.org/10.1007/bf01963532.
  5. R. D’Ambrosio, S. Di Giovacchino, Nonlinear Stability Issues for Stochastic Runge-Kutta Methods, Commun. Nonlinear Sci. Numer. Simul. 94 (2021), 105549. https://doi.org/10.1016/j.cnsns.2020.105549.
  6. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, Berlin, Heidelberg, 1996. https://doi.org/10.1007/978-3-642-05221-7.
  7. S. Anmarkrud, K. Debrabant, A. Kværnø, General Order Conditions for Stochastic Partitioned Runge–Kutta Methods, BIT Numer. Math. 58 (2017), 257–280. https://doi.org/10.1007/s10543-017-0693-6.
  8. E. Platen, N. Bruti-Liberati, Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, Heidelberg, 2010. https://doi.org/10.1007/978-3-642-13694-8.
  9. E. Buckwar, A. Rößler, R. Winkler, Stochastic Runge–Kutta Methods for Itô SODEs with Small Noise, SIAM J. Sci. Comput. 32 (2010), 1789–1808. https://doi.org/10.1137/090763275.
  10. P.M. Burrage, K. Burrage, Structure-Preserving Runge-Kutta Methods for Stochastic Hamiltonian Equations with Additive Noise, Numer. Algorithms 65 (2013), 519–532. https://doi.org/10.1007/s11075-013-9796-6.
  11. K. Burrage, P.M. Burrage, Low Rank Runge–Kutta Methods, Symplecticity and Stochastic Hamiltonian Problems with Additive Noise, J. Comput. Appl. Math. 236 (2012), 3920–3930. https://doi.org/10.1016/j.cam.2012.03.007.
  12. C. Chen, D. Cohen, R. D’Ambrosio, A. Lang, Drift-Preserving Numerical Integrators for Stochastic Hamiltonian Systems, Adv. Comput. Math. 46 (2020), 27. https://doi.org/10.1007/s10444-020-09771-5.
  13. V. Citro, R. D’Ambrosio, S. Di Giovacchino, A-Stability Preserving Perturbation of Runge–Kutta Methods for Stochastic Differential Equations, Appl. Math. Lett. 102 (2020), 106098. https://doi.org/10.1016/j.aml.2019.106098.
  14. R. D’Ambrosio, C. Scalone, On the Numerical Structure Preservation of Nonlinear Damped Stochastic Oscillators, Numer. Algorithms 86 (2020), 933–952. https://doi.org/10.1007/s11075-020-00918-5.
  15. D.J. Higham, X. Mao, A.M. Stuart, Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, SIAM J. Numer. Anal. 40 (2002), 1041–1063. https://doi.org/10.1137/s0036142901389530.
  16. C. Yue, L. Zhao, Strong Convergence of the Split-Step Backward Euler Method for Stochastic Delay Differential Equations with a Nonlinear Diffusion Coefficient, J. Comput. Appl. Math. 382 (2021), 113087. https://doi.org/10.1016/j.cam.2020.113087.
  17. J. Jacod, T. Kurtz, S. Meleard, P. Protter, The Approximate Euler Method for Lévy Driven Stochastic Differential Equations, Ann. Inst. Henri Poincare (B) Probab. Stat. 41 (2005), 523–558. https://doi.org/10.1016/j.anihpb.2004.01.007.
  18. M. Khodabin, K. Maleknejad, M. Rostami, M. Nouri, Numerical Solution of Stochastic Differential Equations by Second Order Runge–Kutta Methods, Math. Comput. Model. 53 (2011), 1910–1920. https://doi.org/10.1016/j.mcm.2011.01.018.
  19. A. Tocino, R. Ardanuy, Runge–Kutta Methods for Numerical Solution of Stochastic Differential Equations, J. Comput. Appl. Math. 138 (2002), 219–241. https://doi.org/10.1016/s0377-0427(01)00380-6.
  20. X. Mao, Stability of Stochastic Differential Equations with Markovian Switching, Stoch. Process. Appl. 79 (1999), 45–67. https://doi.org/10.1016/s0304-4149(98)00070-2.
  21. X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press, 2006. https://doi.org/10.1142/p473.
  22. L. Huang, X. Mao, F. Deng, Stability of Hybrid Stochastic Retarded Systems, IEEE Trans. Circuits Syst. I: Regul. Pap. 55 (2008), 3413–3420. https://doi.org/10.1109/tcsi.2008.2001825.
  23. X. Mao, Stability of Stochastic Differential Equations with Markovian Switching, Stoch. Process. Appl. 79 (1999), 45–67. https://doi.org/10.1016/S0304-4149(98)00070-2.
  24. H. Li, Q. Zhu, The Pth Moment Exponential Stability and Almost Surely Exponential Stability of Stochastic Differential Delay Equations with Poisson Jump, J. Math. Anal. Appl. 471 (2019), 197–210. https://doi.org/10.1016/j.jmaa.2018.10.072.
  25. A.V. Swishchuk, Y.I. Kazmerchuk, Stability of Stochastic Delay Equations of Ito Form With Jumps and Markovian Switchings, and Their Applications in Finance. Theory Probab, Math. Stat. 64 (2002), 167–178.
  26. E. Buckwar, R. D’Ambrosio, Exponential Mean-Square Stability Properties of Stochastic Linear Multistep Methods, Adv. Comput. Math. 47 (2021), 55. https://doi.org/10.1007/s10444-021-09879-2.
  27. R. D’Ambrosio, S.D. Giovacchino, Mean-Square Contractivity of Stochastic $vartheta$-Methods, Commun. Nonlinear Sci. Numer. Simul. 96 (2021), 105671. https://doi.org/10.1016/j.cnsns.2020.105671.
  28. R. D'Ambrosio, C. Scalone, Filon Quadrature for Stochastic Oscillators Driven by Time-Varying Forces, Appl. Numer. Math. 169 (2021), 21–31. https://doi.org/10.1016/j.apnum.2021.06.005.