Mean-square stability of a constructed Third-order stochastic Runge–Kutta schemes for general stochastic differential equations

Farkhonderooz, Omid; Ahmadian, Davood

doi:10.22034/cmde.2021.44264.1871

فهرست نشریات دارای اعتبار وزارت علوم، تحقیقات و فناوری

تعداد نشریات	44
تعداد شماره‌ها	1,303
تعداد مقالات	16,020
تعداد مشاهده مقاله	52,486,968
تعداد دریافت فایل اصل مقاله	15,213,957

	Mean-square stability of a constructed Third-order stochastic Runge–Kutta schemes for general stochastic differential equations
Computational Methods for Differential Equations
مقاله 5، دوره 10، شماره 3، مهر 2022، صفحه 617-638 اصل مقاله (462.44 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22034/cmde.2021.44264.1871
نویسندگان
Omid Farkhonderooz؛ Davood Ahmadian^*
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.
چکیده
In this paper, we are interested in the construction of an explicit third-order stochastic Runge–Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the Itˆo-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered a large number of equations and could find to derive four families for SRK3 schemes. Also, we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.
کلیدواژه‌ها
Stochastic differential equations؛ Stochastic Runge-Kutta schemes؛ Itˆo-Taylor expansion؛ Mean-square stability؛ Convergence

مراجع
[1] D. Ahmadian, O. Farkhondeh Rouz, and L. V. Ballestra, Stability analysis of split-step θ-Milstein method for a class of n-dimensional stochastic differential equations, Appl. Math. Comput., 348(4) (2019), 413–424. [2] D. Ahmadian and O. Farkhondeh Rouz, Exponential mean-square stability of numerical solutions for stochastic delay integro-differential equations with Poisson jump, J. Inequal. Appl., 186(1) (2020), 1–33. [3] L. Arnold, Stochastic differential equations, Applications of Mathematics (New York), Springer-Verlag, Wiley, New York, 1974. [4] K. Burrage and P. M. Burrage, Order conditions of stochastic Runge–Kutta methods by B-series, SIAM J. Numer. Anal., 38(5) (2000), 1626–1646. [5] K. Burrage and P. M. Burrage, High strong order explicit Runge–Kutta methods for stochastic ordinary differential equations, Appl. Numer. Math., 22(1) (1996), 81–101. [6] P. M. Burrage, Runge–Kutta methods for stochastic differential equations, Ph.D. Thesis. The university of queens- land, 1999. [7] J. C. Butcher, The numerical analysis of ordinary differential equations, Runge–Kutta and General Linear Meth- ods, Wiley. Chichester., 1987. [8] E. Buckwar and C. Kelly, Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal., 48(1) (2010), 297–307. [9] C. D. Charalambous, R. J. C. Bultitude, X. Li, and J. Zhan, The estimation of parameters for stochastic differ- ential equations using neural networks, IEEE Trans. Wireless Commun., 15(6) (2008), 434–439. [10] N. T. Dung, Fractional stochastic differential equations with applications to finance, J. Math. Anal. Appl., 397(1) (2013), 334–348. [11] O. Farkhondeh Rouz and D. Ahmadian, Stability analysis of two classes of improved backward Euler methods for stochastic delay differential equations of neutral type, Comput. Methods Differ. Equ., 5(3) (2017), 201–213. [12] O. Farkhondeh Rouz, Preserving asymptotic mean-square stability of stochastic theta scheme for systems of sto- chastic delay differential equations, Comput. Methods Differ. Equ., 8(3) (2020), 468–479. [13] Q. Guo, M. Qiu, and T. Mitsui, Asymptotic mean-square stability of explicit Runge–Kutta Maruyama methods for stochastic delay differential equations, J. Comput. Appl. Math., 296(5) (2015), 1–26. [14] A. Haghighi, S. M. Hosseini, and A. R¨o ler, Diagonally drift-implicit Runge–Kutta methods of strong order one for stiff stochastic differential systems, J. Comput. Appl. Math., 293(2) (2016), 82–93. [15] D. B. Hernandez and R. Spigler, Convergence and stability of implicit Runge–Kutta methods for systems with multiplicative noise, J. Comput. Appl. Math., 33 (1993), 654–669. [16] D. J. Higham and R. Spigler, Mean-square and asymptotic stability of the stochastic theta method, SIAM J. Numer. Anal., 38(3) (2000), 753–769. [17] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Applications of Mathematics (New York), Springer-Verlag, Berlin, 1992. [18] J. R. Klauder and W. P. Petersen, Numerical integration of multiplicative-noise stochastic differential equations, SIAM J. Numer. Anal., 22(6) (1985), 1153–1166. [19] X. Mao, Stochastic Differential Equations and Their Applications, Applications of Mathematics (New York), Springer-Verlag, Horwood. Chichester, 1997. [20] A. Rathinasamy and P. Nair, Asymptotic mean-square stability of weak second-order balanced stochastic Runge– Kutta methods for multi-dimensional Itoˆ stochastic differential systems, Appl. Math. Comput., 332(C) (2018), 276–303. [21] A. R¨o ler, Runge–Kutta methods for Itoˆ stochastic differential equations with scalar noise, BIT., 46(3) (2006), 97–110. [22] A. R¨o ler, Runge–Kutta methods for the strong approximation of solutions of stochastic differential equations, SIAM J. Numer. Anal., 3(1) (2010), 922–952. [23] Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33(6) (1996), 2254–2267. [24] M.J. Senosiain and A. Tocino, Two-step strong order 1.5 schemes for stochastic differential equations, Numer. Algor., DOI:10.1007/s11075-016-0227-3. (2016). [25] A. Tocino, Mean-square stability of second-order Runge–Kutta methods for stochastic differential equations, J. Comput. Appl. Math., 175(3) (2005), 355–367. [26] A. Tocino and R. Ardanuy, Truncated ITOˆ-Taylor expansions, Stochastic Anal. Appl., 20(4) (2002), 427–443. [27] A. Tocino and R. Ardanuy, Runge–Kutta methods for numerical solution of stochastic differential equations, J. Comput. Appl. Math., 138(2) (2002), 219–241. [28] A. Tocino and J. Vigo-Aguiar, Weak second order conditions for stochastic Runge–Kutta methods, SIAM J. Sci. Comput., 24(2) (2002), 507–523. [29] A. Tocino and M. J. Senosiain, Two-step Milstein schemes for stochastic differential equations, Numer. Algor., 69(5) (2014), 643–665. [30] T.H. Tian and K. Burrage, Two-stage stochastic Runge–Kutta methods for stochastic differential equations, BIT., 42(7) (2002), 625–643. [31] P. Wang and Y. Li, Split-step forward methods for stochastic differential equations, J. Comput. Appl. Math., 233(10) (2010), 2641–2651. [32] P. Wilmott, Derivatives: The Theory and Practice of Financial Engineering, Applications of Mathematics (New York), Springer-Verlag, John Wiley & Sons. New York, 1998.
آمار تعداد مشاهده مقاله: 495 تعداد دریافت فایل اصل مقاله: 533

سامانه مدیریت نشریات علمی. قدرت گرفته از سیناوب

پیوندهای مفید

آمار

Mean-square stability of a constructed Third-order stochastic Runge–Kutta schemes for general stochastic differential equations