# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021201
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## The truncated Milstein method for super-linear stochastic differential equations with Markovian switching

 1 Department of Mathematics, Anhui Normal University, Wuhu 241000, China 2 Department of Mathematics, Shanghai Normal University, Shanghai 200234, China 3 Shanghai Customs College, Shanghai 201204, China

* Corresponding author: Qian Guo

Received  September 2019 Revised  October 2020 Early access August 2021

Fund Project: The second author is supported by NSFC of China (No:11871343). The third author is supported by NSFC of China (No:11971303)

In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.

Citation: Weijun Zhan, Qian Guo, Yuhao Cong. The truncated Milstein method for super-linear stochastic differential equations with Markovian switching. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021201
##### References:
 [1] Q. Guo, W. Liu, X. Mao and R. Yue, The truncated Milstein method for stochastic differential equations with commutative noise, J. Comput. Appl. Math., 338 (2018), 298-310.  doi: 10.1016/j.cam.2018.01.014. [2] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015). doi: 10.1090/memo/1112. [3] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803. [4] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence 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.  doi: 10.1098/rspa.2010.0348. [5] X. Li, X. Mao and G. Yin, Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: Truncation methods, convergence in $p$-th moment and stability, IMA J. Numer. Anal., 39 (2019), 847-892.  doi: 10.1093/imanum/dry015. [6] X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.  doi: 10.1016/j.cam.2015.06.002. [7] X. Mao, Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296 (2016), 362-375.  doi: 10.1016/j.cam.2015.09.035. [8] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, 2006.  doi: 10.1142/p473. [9] S. L. Nguyen, T. A. Hoang, D. T. Nguyen and G. Yin, Milstein-type procedures for numerical solutions of stochastic differential equations with Markovian switching, SIAM J. Numer. Anal., 55 (2017), 953-979.  doi: 10.1137/16M1084730. [10] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51 (2013), 3135-3162.  doi: 10.1137/120902318. [11] X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equations Appl., 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617. [12] C. Yuan and X. Mao, Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching, Math. Comput. Simulation, 64 (2004), 223-235.  doi: 10.1016/j.matcom.2003.09.001. [13] S. Zhou, Strong convergence and stability of backward Euler–Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation, Calcolo., 52 (2015), 445-473.  doi: 10.1007/s10092-014-0124-x.

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##### References:
 [1] Q. Guo, W. Liu, X. Mao and R. Yue, The truncated Milstein method for stochastic differential equations with commutative noise, J. Comput. Appl. Math., 338 (2018), 298-310.  doi: 10.1016/j.cam.2018.01.014. [2] M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc., 236 (2015). doi: 10.1090/memo/1112. [3] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients, Ann. Appl. Probab., 22 (2012), 1611-1641.  doi: 10.1214/11-AAP803. [4] M. Hutzenthaler, A. Jentzen and P. E. Kloeden, Strong and weak divergence 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.  doi: 10.1098/rspa.2010.0348. [5] X. Li, X. Mao and G. Yin, Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: Truncation methods, convergence in $p$-th moment and stability, IMA J. Numer. Anal., 39 (2019), 847-892.  doi: 10.1093/imanum/dry015. [6] X. Mao, The truncated Euler–Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 290 (2015), 370-384.  doi: 10.1016/j.cam.2015.06.002. [7] X. Mao, Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations, J. Comput. Appl. Math., 296 (2016), 362-375.  doi: 10.1016/j.cam.2015.09.035. [8] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, 2006.  doi: 10.1142/p473. [9] S. L. Nguyen, T. A. Hoang, D. T. Nguyen and G. Yin, Milstein-type procedures for numerical solutions of stochastic differential equations with Markovian switching, SIAM J. Numer. Anal., 55 (2017), 953-979.  doi: 10.1137/16M1084730. [10] M. V. Tretyakov and Z. Zhang, A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications, SIAM J. Numer. Anal., 51 (2013), 3135-3162.  doi: 10.1137/120902318. [11] X. Wang and S. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Differ. Equations Appl., 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617. [12] C. Yuan and X. Mao, Convergence of the Euler–Maruyama method for stochastic differential equations with Markovian switching, Math. Comput. Simulation, 64 (2004), 223-235.  doi: 10.1016/j.matcom.2003.09.001. [13] S. Zhou, Strong convergence and stability of backward Euler–Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation, Calcolo., 52 (2015), 445-473.  doi: 10.1007/s10092-014-0124-x.
The strong convergence order at the terminal time $T = 1$. The red dashed line is the reference line with the slope of 1
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