Modified Runge–Kutta method with convergence analysis for nonlinear stochastic differential equations with Hölder continuous diffusion coefficient

Iranian Journal of Numerical Analysis and Optimization

دوره 13، شماره 2 - شماره پیاپی 25، شهریور 2023، صفحه 285-316   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.78723.1181

نویسنده

A. Haghighi^*

Department of Mathematics, Faculty of Science, Razi University, Kermanshah 67149, Iran.

چکیده

The main goal of this work is to develop and analyze an accurate trun-cated stochastic Runge–Kutta (TSRK2) method to obtain strong numeri-cal solutions of nonlinear one-dimensional stochastic differential equations (SDEs) with continuous Hölder diffusion coefficients. We will establish the strong L1-convergence theory to the TSRK2 method under the local Lipschitz condition plus the one-sided Lipschitz condition for the drift co-efficient and the continuous Hölder condition for the diffusion coefficient at a time T and over a finite time interval [0, T ], respectively. We show that the new method can achieve the optimal convergence order at a finite time T compared to the classical Euler–Maruyama method. Finally, nu-merical examples are given to support the theoretical results and illustrate the validity of the method.

کلیدواژه‌ها

Stochastic differential equation؛ Strong convergence؛ Truncated methods؛ Hölder continuous coefficient

مراجع

[1] Aït-Sahalia, Y. Transition densities for interest rate and other nonlinear diffusions, J. Fainance 54(4) (1999) 1361–1395.
[2] Buckwar, E. and Kelly, C. Towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations, SIAM J. Numer. Anal. 48(1) (2010), 298–321.
[3] Debrabant, K. and Rößler, A. Diagonally drift-implicit Runge–Kutta methods of weak order one and two for Itô SDEs and stability analysis, Appl. Numer. Math. 59(3) (2009) 595–607.
[4] Emmanuel, C. and Mao, X. Truncated EM numerical method for gener-alised Aït-Sahalia-type interest rate model with delay, J. Comput. Appl. Math. 383 (2021), 113137.

[5] Falsone, G. Stochastic differential calculus for Gaussian and non-Gaussian noises: A critical review, Commum. Nonlinear. Sci. 56 (2018), 198–216.
[6] Fatemion Aghda, A.S., Hosseini, S.M. and Tahmasebi, M. Analysis of non-negativity and convergence of solution of the balanced implicit method for the delay Cox–Ingersoll–Ross model, Appl. Numer. Math. 118 (2017), 249–265.
[7] Greenhalgh, D., Liang Y., and Mao, X. Demographic stochasticity in the SDE SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B. 20(9) (2015) 2859–2884.
[8] Haghighi, A. New S-ROCK methods for stochastic differential equations with commutative noise, Iran. J. Numer. Anal. Optim. 9(1) (2019), 105–126.
[9] Haghighi, A. and Rößler, A. Split-step double balanced approximation methods for stiff stochastic differential equations, Int. J. Appl. Math. 96(5) (2019), 1030–1047.
[10] Higham, D.J., Mao, X. and Stuart, A. M. Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal. 40(3) (2002) 1041–1063.
[11] Hu, L., Li, X. and Mao, X. Convergence rate and stability of the trun-cated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 337 (2018) 274–289.
[12] Hutzenthaler, M. and Jentzen, A. Numerical approximations of stochas-tic differential equations with non-globally Lipschitz continuous coeffi-cients, Providence, RI: American Mathematical Society (AMS), 2015.
[13] Hutzenthaler, M., Jentzen, A. and Kloeden, P.E. Strong and weak diver-gence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci 467 (2011) 1563–1576.
[14] Hutzenthaler, M., Jentzen, A. and Kloeden, P.E. Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients, Ann. Appl. Probab. 22(4) (2012) 1611–1641.
[15] Ikeda, N. and Watanabe, S. Stochastic Differential Equations and Dif-fusion Process, North-Holand, Amesterdam, 1981.
[16] Kloeden, P. and Platen, E. Numerical solution of stochastic differential equations, Vol. 23, Springer-Verlag, Berlin, 1999.

[17] Komori, Y. and Burrage, K. Strong first order S-ROCK methods for stochastic differential equations, J. Comput. Appl. Math. 242 (2013), 261–274.
[18] Li, X., Mao X. and Yin, G. Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: trunca-tion methods, convergence in pth moment and stability, IMA J. Numer. Anal. 39(2) (2018), 847–892.
[19] Liu, W. and Mao, X. Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, Appl. Math. Com-put. 223 (2013), 389–400.
[20] Mao, X. Stochastic Differential Equations and Applications, 2nd edition, Horwood, Chichester, UK, 2007.
[21] Mao, X. The truncated Euler–Maruyama method for stochastic differen-tial equations, J. Comput. Appl. Math. 290 (2015) 370–384.
[22] Mao, X. Convergence rates of the truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math. 296 (2016) 362–375.
[23] Mao, X. and Szpruch, L. Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math. 238 (2013) 14–28.
[24] Milstein, G. N. Numerical Integration of Stochastic Differential Equa-tions, Kluwer Academic, Dordrecht, 1995.
[25] Milstein, G.N., Platen, E. and Schurz, H. Balanced implicit methods for stiff stochastic system, SIAM J. Numer. Anal. 35(3) (1998) 1010–1019.
[26] Øksendal, B. Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, Berlin, 2003.
[27] Rößler, A. Runge–Kutta methods for the strong approximation of solu-tions of stochastic differential equations, SIAM J. Numer. Anal. 48(3)(2010) 922–952.
[28] Sabanis, S. A note on tamed Euler approximations, Electron. Comm. Probab. 18(47) (2013) 1–10.
[29] Soheili, R.A., Amini, M. and Soleymani, F. A family of Chaplygin-type solvers for Itô stochastic differential equations, Appl. Math. Comput. 340 (2019) 296–304.
[30] Tretyakov, M. and Zhang, Z. A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal. 51(6) (2013) 3135–3162.

[31] Yamada, T. and Watanabe, S. On the uniqueness of stochastic differen-tial equations, J. Math. Kyoto Univ. 11 (1971) 155–167.
[32] Yang, H., Fuke, W., Kloeden P.E. and Mao, X. The truncated Euler–Maruyama method for stochastic differential equations with Hölder dif-fusion coefficients, J. Comput. Appl. Math. 366 (2020) 112379.
[33] Yang, H. and Huang, J. Convergence and stability of modified partially truncated Euler–Maruyama method for nonlinear stochastic differential equations with Hölder continuous diffusion coefficient, J. Comput. Appl. Math. 404 (2022) 113895.