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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July
2022, 27(7): 3663-3682.
doi: 10.3934/dcdsb.2021201
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 |
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 and Continuous Dynamical Systems - B,
2022, 27
(7)
: 3663-3682.
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. |
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. |
| [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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