February  2019, 24(2): 695-717. doi: 10.3934/dcdsb.2018203

Convergence rate and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments

Department of Mathematics, Harbin Institute of Technology, Harbin, China, 150001

* Corresponding author: Minghui Song

Received  July 2017 Revised  March 2018 Published  February 2019 Early access  June 2018

Fund Project: This work is supported by the NSF of P.R. China (No.11671113).

In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with $θ∈[\frac{1}{2},1]$ is strongly convergent with order $\frac{1}{2}$ in $p$th($p≥ 2$) moment if both drift and diffusion coefficients are polynomially growing with regard to the delay terms, while the diffusion coefficients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with $θ∈(\frac{1}{2},1]$ is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists $θ^*∈(\frac{1}{2},1]$ such that the ISST method with $θ∈(θ^*,1]$ is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.

Citation: 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 and Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203
References:
[1]

J. H. Bao and C. G. Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc., 141 (2013), 3231-3243.  doi: 10.1090/S0002-9939-2013-11886-1.

[2]

W. J. BeynE. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput., 67 (2016), 955-987.  doi: 10.1007/s10915-015-0114-4.

[3]

W. J. BeynE. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput., 70 (2017), 1042-1077.  doi: 10.1007/s10915-016-0290-x.

[4]

K. DareiotisC. Kumar and S. Sabanis, On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations, SIAM J. Numer. Anal., 54 (2016), 1840-1872.  doi: 10.1137/151004872.

[5]

Q. GuoW. LiuX. R. Mao and R. X. Yue, The partially truncated Euler-Maruyama method and its stability and boundedness, Appl. Numer. Math., 115 (2017), 235-251.  doi: 10.1016/j.apnum.2017.01.010.

[6]

D. J. HighamX. R. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.  doi: 10.1137/S0036142901389530.

[7]

Y. Z. Hu, Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, Progr. Probab., 38 (1996), 183-202. 

[8]

C. M. Huang, Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math., 236 (2012), 4016-4026.  doi: 10.1016/j.cam.2012.03.005.

[9]

C. M. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math., 259 (2014), 77-86.  doi: 10.1016/j.cam.2013.03.038.

[10]

M. HutzenthalerA. 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.

[11]

M. HutzenthalerA. 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.

[12]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[13]

C. Kumar and S. Sabanis, Strong convergence of Euler approximations of stochastic differential equations with delay under local Lipschitz condition, Stoch. Anal. Appl., 32 (2014), 207-228.  doi: 10.1080/07362994.2014.858552.

[14]

W. Liu and X. R. Mao, Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, J. Comput. Appl. Math., 223 (2013), 389-400.  doi: 10.1016/j.amc.2013.08.023.

[15]

Y. L. LuM. H. Song and M. Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math., 317 (2017), 55-71.  doi: 10.1016/j.cam.2016.11.033.

[16]

X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.

[17]

X. R. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85 (2013), 144-171.  doi: 10.1080/17442508.2011.651213.

[18]

X. R. MaoW. LiuL. J. HuQ. Luo and J. Q. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.  doi: 10.1016/j.sysconle.2014.08.011.

[19]

X. R. 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.

[20]

X. R. 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.

[21]

M. Milošević, The Euler-Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1-12.  doi: 10.1016/j.cam.2015.11.019.

[22]

M. Milošević, Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay, J. Comput. Appl. Math., 280 (2015), 248-264.  doi: 10.1016/j.cam.2014.12.002.

[23]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.

[24]

M. H. Song and L. Zhang, Numerical solutions of stochastic differential equations with piecewise continuous arguments under Khasminskii-Type conditions, J. Appl. Math., 2012 (2012), Art. ID 696849, 21 pp.

[25]

M. V. Tretyakov and Z. Q. 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.

[26]

X. J. Wang and S. Q. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Difference Equ. Appl., 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617.

[27]

F. K. WuX. R. Mao and K. Chen, The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Appl. Numer. Math., 59 (2009), 2641-2658.  doi: 10.1016/j.apnum.2009.03.004.

[28]

S. R. YouW. LiuJ. Q. LuX. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim., 53 (2015), 905-925.  doi: 10.1137/140985779.

[29]

L. Zhang and M. H. Song, Convergence of the Euler method of stochastic differential equations with piecewise continuous arguments, Abstr. Appl. Anal., 2012 (2012), Art. ID 643783, 16 pp.

[30]

S. B. 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.

[31]

X. F. ZongF. K. Wu and C. M. Huang, Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 278 (2015), 258-277.  doi: 10.1016/j.cam.2014.10.014.

show all references

References:
[1]

J. H. Bao and C. G. Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc., 141 (2013), 3231-3243.  doi: 10.1090/S0002-9939-2013-11886-1.

[2]

W. J. BeynE. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes, J. Sci. Comput., 67 (2016), 955-987.  doi: 10.1007/s10915-015-0114-4.

[3]

W. J. BeynE. Isaak and R. Kruse, Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes, J. Sci. Comput., 70 (2017), 1042-1077.  doi: 10.1007/s10915-016-0290-x.

[4]

K. DareiotisC. Kumar and S. Sabanis, On tamed Euler approximations of SDEs driven by Lévy noise with applications to delay equations, SIAM J. Numer. Anal., 54 (2016), 1840-1872.  doi: 10.1137/151004872.

[5]

Q. GuoW. LiuX. R. Mao and R. X. Yue, The partially truncated Euler-Maruyama method and its stability and boundedness, Appl. Numer. Math., 115 (2017), 235-251.  doi: 10.1016/j.apnum.2017.01.010.

[6]

D. J. HighamX. R. Mao and A. M. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2002), 1041-1063.  doi: 10.1137/S0036142901389530.

[7]

Y. Z. Hu, Semi-implicit Euler-Maruyama scheme for stiff stochastic equations, Progr. Probab., 38 (1996), 183-202. 

[8]

C. M. Huang, Exponential mean square stability of numerical methods for systems of stochastic differential equations, J. Comput. Appl. Math., 236 (2012), 4016-4026.  doi: 10.1016/j.cam.2012.03.005.

[9]

C. M. Huang, Mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations, J. Comput. Appl. Math., 259 (2014), 77-86.  doi: 10.1016/j.cam.2013.03.038.

[10]

M. HutzenthalerA. 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.

[11]

M. HutzenthalerA. 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.

[12]

P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer, Berlin, 1992. doi: 10.1007/978-3-662-12616-5.

[13]

C. Kumar and S. Sabanis, Strong convergence of Euler approximations of stochastic differential equations with delay under local Lipschitz condition, Stoch. Anal. Appl., 32 (2014), 207-228.  doi: 10.1080/07362994.2014.858552.

[14]

W. Liu and X. R. Mao, Strong convergence of the stopped Euler-Maruyama method for nonlinear stochastic differential equations, J. Comput. Appl. Math., 223 (2013), 389-400.  doi: 10.1016/j.amc.2013.08.023.

[15]

Y. L. LuM. H. Song and M. Z. Liu, Convergence and stability of the split-step theta method for stochastic differential equations with piecewise continuous arguments, J. Comput. Appl. Math., 317 (2017), 55-71.  doi: 10.1016/j.cam.2016.11.033.

[16]

X. R. Mao, Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control, Automatica J. IFAC, 49 (2013), 3677-3681.  doi: 10.1016/j.automatica.2013.09.005.

[17]

X. R. Mao and L. Szpruch, Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients, Stochastics, 85 (2013), 144-171.  doi: 10.1080/17442508.2011.651213.

[18]

X. R. MaoW. LiuL. J. HuQ. Luo and J. Q. Lu, Stabilization of hybrid stochastic differential equations by feedback control based on discrete-time state observations, Systems Control Lett., 73 (2014), 88-95.  doi: 10.1016/j.sysconle.2014.08.011.

[19]

X. R. 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.

[20]

X. R. 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.

[21]

M. Milošević, The Euler-Maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments, J. Comput. Appl. Math., 298 (2016), 1-12.  doi: 10.1016/j.cam.2015.11.019.

[22]

M. Milošević, Convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay, J. Comput. Appl. Math., 280 (2015), 248-264.  doi: 10.1016/j.cam.2014.12.002.

[23]

Y. Saito and T. Mitsui, Stability analysis of numerical schemes for stochastic differential equations, SIAM J. Numer. Anal., 33 (1996), 2254-2267.  doi: 10.1137/S0036142992228409.

[24]

M. H. Song and L. Zhang, Numerical solutions of stochastic differential equations with piecewise continuous arguments under Khasminskii-Type conditions, J. Appl. Math., 2012 (2012), Art. ID 696849, 21 pp.

[25]

M. V. Tretyakov and Z. Q. 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.

[26]

X. J. Wang and S. Q. Gan, The tamed Milstein method for commutative stochastic differential equations with non-globally Lipschitz continuous coefficients, J. Difference Equ. Appl., 19 (2013), 466-490.  doi: 10.1080/10236198.2012.656617.

[27]

F. K. WuX. R. Mao and K. Chen, The Cox-Ingersoll-Ross model with delay and strong convergence of its Euler-Maruyama approximate solutions, Appl. Numer. Math., 59 (2009), 2641-2658.  doi: 10.1016/j.apnum.2009.03.004.

[28]

S. R. YouW. LiuJ. Q. LuX. R. Mao and Q. W. Qiu, Stabilization of hybrid systems by feedback control based on discrete-time state observations, SIAM J. Control Optim., 53 (2015), 905-925.  doi: 10.1137/140985779.

[29]

L. Zhang and M. H. Song, Convergence of the Euler method of stochastic differential equations with piecewise continuous arguments, Abstr. Appl. Anal., 2012 (2012), Art. ID 643783, 16 pp.

[30]

S. B. 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.

[31]

X. F. ZongF. K. Wu and C. M. Huang, Theta schemes for SDDEs with non-globally Lipschitz continuous coefficients, J. Comput. Appl. Math., 278 (2015), 258-277.  doi: 10.1016/j.cam.2014.10.014.

Figure 1.  (a) The mean square errors. (b) The 3th moment errors
Figure 2.  (a) The mean square errors. (b) The 3th moment errors
Figure 3.  (a) $a = -3,~b = 0,~c = 1$. (b) $a = -1.8,~b = 0.4,~c = 0.7$
Table 1.  Mean square errors $\mathbb{E}|x(5)-x_{5m}|^2$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(5)$ rate $\epsilon(5)$ rate $\epsilon(5)$ rate
$2^{-6}$ $0.2330e-04$ $ * $ $ 0.2121e-04$ $ * $ $0.1974e-04$ $ * $
$2^{-7}$ $ 0.1037e-04$ $ 2.2469 $ $0.0982e-04$ $ 2.1599$ $0.0943e-04$ $2.0933$
$2^{-8}$ $0.0444e-04$ $ 2.3356 $ $0.0435e-04$ $ 2.2575$ $ 0.0414e-04$ $2.2778$
$2^{-9}$ $0.0184e-04$ $ 2.4130 $ $0.0179e-04$ $2.4302$ $0.0175e-04$ $ 2.3657$
$2^{-10}$ $0.0096e-04$ $1.9167$ $0.0096e-04$ $1.8646$ $0.0096e-04$ $1.8229$
$2^{-11}$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(5)$ rate $\epsilon(5)$ rate $\epsilon(5)$ rate
$2^{-6}$ $0.2330e-04$ $ * $ $ 0.2121e-04$ $ * $ $0.1974e-04$ $ * $
$2^{-7}$ $ 0.1037e-04$ $ 2.2469 $ $0.0982e-04$ $ 2.1599$ $0.0943e-04$ $2.0933$
$2^{-8}$ $0.0444e-04$ $ 2.3356 $ $0.0435e-04$ $ 2.2575$ $ 0.0414e-04$ $2.2778$
$2^{-9}$ $0.0184e-04$ $ 2.4130 $ $0.0179e-04$ $2.4302$ $0.0175e-04$ $ 2.3657$
$2^{-10}$ $0.0096e-04$ $1.9167$ $0.0096e-04$ $1.8646$ $0.0096e-04$ $1.8229$
$2^{-11}$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$
Table 2.  The $3$th moment errors $\mathbb{E}|x(5)-x_{5m}|^3$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(5)$ rate $\epsilon(5)$ rate $\epsilon(5)$ rate
$2^{-6}$ $0.4083e-06$ $ * $ $ 0.4032e-06$ $ * $ $0.4039e-06$ $ * $
$2^{-7}$ $ 0.1266e-06$ $ 3.2251 $ $0.1245e-06$ $ 3.2386$ $0.1203e-06$ $ 3.3574$
$2^{-8}$ $0.0373e-06$ $ 3.3941 $ $0.0364e-06$ $3.4203$ $ 0.0352e-06$ $3.4176$
$2^{-9}$ $0.0127e-06$ $ 2.9370 $ $0.0115e-06$ $3.1652$ $0.0093e-06$ $3.7849$
$2^{-10}$ $0.0067e-06$ $ 1.8955 $ $0.0055e-06$ $2.0909$ $0.0049e-06$ $ 1.8980$
$2^{-11}$ $0.0011e-06$ $ 6.0909 $ $0.0011e-06$ $5.0000$ $0.0011e-06$ $ 4.4545$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(5)$ rate $\epsilon(5)$ rate $\epsilon(5)$ rate
$2^{-6}$ $0.4083e-06$ $ * $ $ 0.4032e-06$ $ * $ $0.4039e-06$ $ * $
$2^{-7}$ $ 0.1266e-06$ $ 3.2251 $ $0.1245e-06$ $ 3.2386$ $0.1203e-06$ $ 3.3574$
$2^{-8}$ $0.0373e-06$ $ 3.3941 $ $0.0364e-06$ $3.4203$ $ 0.0352e-06$ $3.4176$
$2^{-9}$ $0.0127e-06$ $ 2.9370 $ $0.0115e-06$ $3.1652$ $0.0093e-06$ $3.7849$
$2^{-10}$ $0.0067e-06$ $ 1.8955 $ $0.0055e-06$ $2.0909$ $0.0049e-06$ $ 1.8980$
$2^{-11}$ $0.0011e-06$ $ 6.0909 $ $0.0011e-06$ $5.0000$ $0.0011e-06$ $ 4.4545$
Table 3.  Mean square errors $\mathbb{E}|x(3)-x_{3m}|^2$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(3)$ rate $\epsilon(3)$ rate $\epsilon(3)$ rate
$2^{-6}$ $1.0839e-04$ $ * $ $ 1.0689e-03$ $ * $ $1.0578e-03$ $ * $
$2^{-7}$ $5.1071e-04$ $2.1223$ $5.0147e-04$ $ 2.1315$ $4.9992e-04 $ $ 2.1159 $
$2^{-8}$ $ 2.6099e-04$ $1.9568$ $2.5515e-04$ $ 2.1000$ $2.5056e-04$ $ 1.9952 $
$2^{-9}$ $1.2395e-04$ $ 2.1056$ $1.2150e-04$ $1.9158$ $1.1603e-04 $ $ 2.1592 $
$2^{-10}$ $0.6654e-04$ $ 1.8628$ $0.6342e-04$ $1.8646$ $0.5864e-04$ $ 1.9787$
$2^{-11}$ $0.3179e-04$ $2.0931$ $0.3155e-04$ $2.0101$ $0.3124e-04$ $ 1.8770$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(3)$ rate $\epsilon(3)$ rate $\epsilon(3)$ rate
$2^{-6}$ $1.0839e-04$ $ * $ $ 1.0689e-03$ $ * $ $1.0578e-03$ $ * $
$2^{-7}$ $5.1071e-04$ $2.1223$ $5.0147e-04$ $ 2.1315$ $4.9992e-04 $ $ 2.1159 $
$2^{-8}$ $ 2.6099e-04$ $1.9568$ $2.5515e-04$ $ 2.1000$ $2.5056e-04$ $ 1.9952 $
$2^{-9}$ $1.2395e-04$ $ 2.1056$ $1.2150e-04$ $1.9158$ $1.1603e-04 $ $ 2.1592 $
$2^{-10}$ $0.6654e-04$ $ 1.8628$ $0.6342e-04$ $1.8646$ $0.5864e-04$ $ 1.9787$
$2^{-11}$ $0.3179e-04$ $2.0931$ $0.3155e-04$ $2.0101$ $0.3124e-04$ $ 1.8770$
Table 4.  The $3$th moment errors $\mathbb{E}|x(3)-x_{3m}|^3$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(3)$ rate $\epsilon(3)$ rate $\epsilon(3)$ rate
$2^{-6}$ $3.4673e-05$ $ * $ $ 3.4000e-05$ $ * $ $3.3598e-05$ $ * $
$2^{-7}$ $1.0647e-05 $ $ 3.2566 $ $1.0435e-05$ $ 3.2583$ $1.0150e-05$ $3.3101$
$2^{-8}$ $3.3059e-06$ $ 3.2206 $ $3.2294e-06$ $ 3.2313$ $ 3.0933e-06$ $3.2813$
$2^{-9}$ $1.0265e-06 $ $ 3.2206 $ $1.0258e-06$ $3.1481$ $0.9983e-06$ $ 3.0986$
$2^{-10}$ $0.3531e-06$ $ 2.9071$ $0.3518e-06$ $2.9159$ $0.3426e-06$ $ 2.9139$
$2^{-11}$ $0.1560e-06$ $ 2.2635$ $0.1556e-06$ $2.2609$ $0.1515e-06$ $2.2614$
step size $h$ $\theta=0.5$ $\theta=0.75$ $\theta=1$
$\epsilon(3)$ rate $\epsilon(3)$ rate $\epsilon(3)$ rate
$2^{-6}$ $3.4673e-05$ $ * $ $ 3.4000e-05$ $ * $ $3.3598e-05$ $ * $
$2^{-7}$ $1.0647e-05 $ $ 3.2566 $ $1.0435e-05$ $ 3.2583$ $1.0150e-05$ $3.3101$
$2^{-8}$ $3.3059e-06$ $ 3.2206 $ $3.2294e-06$ $ 3.2313$ $ 3.0933e-06$ $3.2813$
$2^{-9}$ $1.0265e-06 $ $ 3.2206 $ $1.0258e-06$ $3.1481$ $0.9983e-06$ $ 3.0986$
$2^{-10}$ $0.3531e-06$ $ 2.9071$ $0.3518e-06$ $2.9159$ $0.3426e-06$ $ 2.9139$
$2^{-11}$ $0.1560e-06$ $ 2.2635$ $0.1556e-06$ $2.2609$ $0.1515e-06$ $2.2614$
[1]

Wolf-Jürgen Beyn, Raphael Kruse. Two-sided error estimates for the stochastic theta method. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 389-407. doi: 10.3934/dcdsb.2010.14.389

[2]

Weidong Zhao, Jinlei Wang, Shige Peng. Error estimates of the $\theta$-scheme for backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 905-924. doi: 10.3934/dcdsb.2009.12.905

[3]

Weidong Zhao, Yang Li, Guannan Zhang. A generalized $\theta$-scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

[4]

Pham Huu Anh Ngoc. New criteria for exponential stability in mean square of stochastic functional differential equations with infinite delay. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021040

[5]

Abdulkarim Hassan Ibrahim, Poom Kumam, Min Sun, Parin Chaipunya, Auwal Bala Abubakar. Projection method with inertial step for nonlinear equations: Application to signal recovery. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021173

[6]

Chuchu Chen, Jialin Hong. Mean-square convergence of numerical approximations for a class of backward stochastic differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (8) : 2051-2067. doi: 10.3934/dcdsb.2013.18.2051

[7]

Nora Merabet. Global convergence of a memory gradient method with closed-form step size formula. Conference Publications, 2007, 2007 (Special) : 721-730. doi: 10.3934/proc.2007.2007.721

[8]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521

[9]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete and Continuous Dynamical Systems, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324

[10]

Zhen Li, Jicheng Liu. Synchronization for stochastic differential equations with nonlinear multiplicative noise in the mean square sense. Discrete and Continuous Dynamical Systems - B, 2019, 24 (10) : 5709-5736. doi: 10.3934/dcdsb.2019103

[11]

Hailong Zhu, Jifeng Chu, Weinian Zhang. Mean-square almost automorphic solutions for stochastic differential equations with hyperbolicity. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1935-1953. doi: 10.3934/dcds.2018078

[12]

Wei Wang, Kai Liu, Xiulian Wang. Sensitivity to small delays of mean square stability for stochastic neutral evolution equations. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2403-2418. doi: 10.3934/cpaa.2020105

[13]

Quan Zhou, Yabing Sun. High order one-step methods for backward stochastic differential equations via Itô-Taylor expansion. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021233

[14]

Jamilu Abubakar, Poom Kumam, Abor Isa Garba, Muhammad Sirajo Abdullahi, Abdulkarim Hassan Ibrahim, Wachirapong Jirakitpuwapat. An efficient iterative method for solving split variational inclusion problem with applications. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021160

[15]

Zeng-Zhen Tan, Rong Hu, Ming Zhu, Ya-Ping Fang. A dynamical system method for solving the split convex feasibility problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 2989-3011. doi: 10.3934/jimo.2020104

[16]

Preeyanuch Chuasuk, Ferdinard Ogbuisi, Yekini Shehu, Prasit Cholamjiak. New inertial method for generalized split variational inclusion problems. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3357-3371. doi: 10.3934/jimo.2020123

[17]

Chibueze Christian Okeke, Abdulmalik Usman Bello, Lateef Olakunle Jolaoso, Kingsley Chimuanya Ukandu. Inertial method for split null point problems with pseudomonotone variational inequality problems. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021037

[18]

Behrouz Kheirfam, Guoqiang Wang. An infeasible full NT-step interior point method for circular optimization. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 171-184. doi: 10.3934/naco.2017011

[19]

Van Hieu Dang. An extension of hybrid method without extrapolation step to equilibrium problems. Journal of Industrial and Management Optimization, 2017, 13 (4) : 1723-1741. doi: 10.3934/jimo.2017015

[20]

Jingwen Wu, Jintao Hu, Hongjiong Tian. Functionally-fitted block $ \theta $-methods for ordinary differential equations. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2603-2617. doi: 10.3934/dcdss.2020164

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (357)
  • HTML views (472)
  • Cited by (1)

Other articles
by authors

[Back to Top]