Truncated Milstein method for non-autonomous stochastic differential equations and its modification

The truncated Milstein method, which was initial proposed in (Guo, Liu, Mao and Yue 2018), is extended to the non-autonomous stochastic differential equations with the super-linear state variable and the Holder continuous time variable. The convergence rate is proved. Compared with the initial work, the requirements on the step-size is significantly released. In addition, the technique of the randomized step-size is employed to raise the convergence rate of the truncated Milstein method.

[1]  Mingzhu Liu,et al.  Convergence of the Tamed Euler Method for Stochastic Differential Equations with Piecewise Continuous Arguments Under Non-global Lipschitz Continuous Coefficients , 2018 .

[2]  P. Kloeden,et al.  Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients , 2009, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[3]  Jessica Fuerst,et al.  Stochastic Differential Equations And Applications , 2016 .

[4]  E. Hairer,et al.  Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems , 2010 .

[5]  A. Xiao,et al.  Strong convergence of the split-step theta method for neutral stochastic delay differential equations ☆ , 2017 .

[6]  Daniel E. Geer,et al.  Convergence , 2021, IEEE Secur. Priv..

[7]  Tianhai Tian,et al.  Implicit Stochastic Runge–Kutta Methods for Stochastic Differential Equations , 2004 .

[8]  R. Kruse,et al.  A randomized Milstein method for stochastic differential equations with non-differentiable drift coefficients , 2017, Discrete & Continuous Dynamical Systems - B.

[9]  Xuerong Mao,et al.  Explicit numerical approximations for stochastic differential equations in finite and infinite horizons: truncation methods, convergence in pth moment, and stability , 2019 .

[10]  D. Higham.  Stochastic Ordinary Differential Equations in Applied and Computational Mathematics , 2011 .

[11]  Liangjian Hu,et al.  Convergence rate and stability of the truncated Euler-Maruyama method for stochastic differential equations , 2018, J. Comput. Appl. Math..

[12]  Xiaojie Wang,et al.  Tamed Runge-Kutta methods for SDEs with super-linearly growing drift and diffusion coefficients , 2020, Applied Numerical Mathematics.

[13]  Wei Liu,et al.  The truncated EM method for stochastic differential equations with Poisson jumps , 2018, J. Comput. Appl. Math..

[14]  Boleslaw Z. Kacewicz,et al.  Optimal solution of ordinary differential equations , 1987, J. Complex..

[15]  Heping Ma,et al.  Order-preserving strong schemes for SDEs with locally Lipschitz coefficients , 2014, 1402.3708.

[16]  F. Krogh,et al.  Solving Ordinary Differential Equations , 2019, Programming for Computations - Python.

[17]  Wei Liu,et al.  The truncated Milstein method for stochastic differential equations with commutative noise , 2017, J. Comput. Appl. Math..

[18]  Xiaojie Wang,et al.  The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients , 2011, 1102.0662.

[19]  Xiaojie Wang,et al.  Mean-square convergence rates of stochastic theta methods for SDEs under a coupled monotonicity condition , 2020 .

[20]  Wolf-Jürgen Beyn,et al.  Two-Sided Error Estimates for the Stochastic Theta Method , 2010 .

[21]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[22]  W. Liu,et al.  Equivalence of the mean square stability between the partially truncated Euler–Maruyama method and stochastic differential equations with super-linear growing coefficients , 2018, Advances in Difference Equations.

[23]  Xuerong Mao,et al.  Convergence rates of the truncated Euler-Maruyama method for stochastic differential equations , 2016, J. Comput. Appl. Math..

[24]  Xuerong Mao,et al.  Strong convergence rates for backward Euler–Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients , 2013 .

[25]  Thomas Daun.  On the randomized solution of initial value problems , 2011, J. Complex..

[26]  L. Szpruch,et al.  Convergence, Non-negativity and Stability of a New Milstein Scheme with Applications to Finance , 2012, 1204.1647.

[27]  Konstantinos Dareiotis,et al.  On Tamed Euler Approximations of SDEs Driven by Lévy Noise with Applications to Delay Equations , 2014, SIAM J. Numer. Anal..

[28]  P. Kloeden,et al.  Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients , 2010, 1010.3756.

[29]  Chengming Huang,et al.  Convergence and stability of the semi-tamed Euler scheme for stochastic differential equations with non-Lipschitz continuous coefficients , 2014, Appl. Math. Comput..

[30]  Guangqiang Lan,et al.  Strong convergence rates of modified truncated EM method for stochastic differential equations , 2017, J. Comput. Appl. Math..

[31]  A. Khaliq,et al.  Split-step Milstein methods for multi-channel stiff stochastic differential systems , 2014, 1411.7080.

[32]  Kristian Debrabant,et al.  Diagonally drift-implicit Runge--Kutta methods of weak order one and two for Itô SDEs and stability analysis , 2009 .

[33]  Christian Kahl,et al.  Balanced Milstein Methods for Ordinary SDEs , 2006, Monte Carlo Methods Appl..

[34]  Xuerong Mao,et al.  The partially truncated Euler–Maruyama method and its stability and boundedness , 2017 .

[35]  Wei Zhang,et al.  Strong convergence of the partially truncated Euler-Maruyama method for a class of stochastic differential delay equations , 2018, J. Comput. Appl. Math..

[36]  Xuerong Mao,et al.  The truncated Euler-Maruyama method for stochastic differential equations , 2015, J. Comput. Appl. Math..

[37]  Yaozhong Hu.  Semi-Implicit Euler-Maruyama Scheme for Stiff Stochastic Equations , 1996 .

[38]  X. Mao,et al.  Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations , 2018, Applied Numerical Mathematics.

[39]  S. Hosseini,et al.  Stochastic RungeKutta Rosenbrock type methods for SDE systems , 2017 .

[40]  H. Ngo,et al.  Tamed Euler–Maruyama approximation for stochastic differential equations with locally Hölder continuous diffusion coefficients , 2019, Statistics & Probability Letters.

[41]  Fuke Wu,et al.  Convergence and stability of two classes of theta-Milstein schemes for stochastic differential equations , 2015, J. Comput. Appl. Math..