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

Weijun Zhan 1, , Qian Guo 2,, and  Yuhao Cong 3,

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.

Keywords: Stochastic differential equation, Milstein method, Khasminskii-type condition, strong convergence, Markovian switching.
Mathematics Subject Classification: Primary: 60H35; Secondary: 65C30.
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

Figure 1.  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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