# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021201
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## 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 & 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, 2006.  doi: 10.1142/p473.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

show all references

##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] X. Mao and C. Yuan, Stochastic Differential Equations with Markovian Switching,, Imperial College Press, 2006.  doi: 10.1142/p473.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
The strong convergence order at the terminal time $T = 1$. The red dashed line is the reference line with the slope of 1
 [1] Minghui Song, Liangjian Hu, Xuerong Mao, Liguo Zhang. Khasminskii-type theorems for stochastic functional differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1697-1714. doi: 10.3934/dcdsb.2013.18.1697 [2] Yaozhong Hu, David Nualart, Xiaobin Sun, Yingchao Xie. Smoothness of density for stochastic differential equations with Markovian switching. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3615-3631. doi: 10.3934/dcdsb.2018307 [3] Raphael Kruse, Yue Wu. A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3475-3502. doi: 10.3934/dcdsb.2018253 [4] Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5567-5579. doi: 10.3934/dcdsb.2020367 [5] Fuke Wu, George Yin, Le Yi Wang. Razumikhin-type theorems on moment exponential stability of functional differential equations involving two-time-scale Markovian switching. Mathematical Control & Related Fields, 2015, 5 (3) : 697-719. doi: 10.3934/mcrf.2015.5.697 [6] Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4887-4905. doi: 10.3934/dcdsb.2020317 [7] Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108 [8] Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287 [9] Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, 2020, 10 (2) : 443-454. doi: 10.3934/mcrf.2020005 [10] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [11] Xiaojin Huang, Hongfu Yang, Jianhua Huang. Consensus stability analysis for stochastic multi-agent systems with multiplicative measurement noises and Markovian switching topologies. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021024 [12] Yulan Lu, Minghui Song, Mingzhu Liu. Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203 [13] Muslim Malik, Anjali Rose, Anil Kumar. Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021068 [14] Liu Liu. Uniform spectral convergence of the stochastic Galerkin method for the linear semiconductor Boltzmann equation with random inputs and diffusive scaling. Kinetic & Related Models, 2018, 11 (5) : 1139-1156. doi: 10.3934/krm.2018044 [15] Fuke Wu, Xuerong Mao, Peter E. Kloeden. Discrete Razumikhin-type technique and stability of the Euler--Maruyama method to stochastic functional differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885 [16] Weijun Zhou, Youhua Zhou. On the strong convergence of a modified Hestenes-Stiefel method for nonconvex optimization. Journal of Industrial & Management Optimization, 2013, 9 (4) : 893-899. doi: 10.3934/jimo.2013.9.893 [17] Ying Hu, Shanjian Tang. Switching game of backward stochastic differential equations and associated system of obliquely reflected backward stochastic differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5447-5465. doi: 10.3934/dcds.2015.35.5447 [18] Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280 [19] Desmond J. Higham, Xuerong Mao, Lukasz Szpruch. Convergence, non-negativity and stability of a new Milstein scheme with applications to finance. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2083-2100. doi: 10.3934/dcdsb.2013.18.2083 [20] Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285

2020 Impact Factor: 1.327