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  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 & 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 & 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 & 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 & Continuous Dynamical Systems - B, 2012, 17 (5) : 1585-1603. doi: 10.3934/dcdsb.2012.17.1585

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

Marat Akhmet, Duygu Aruğaslan. Lyapunov-Razumikhin method for differential equations with piecewise constant argument. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 457-466. doi: 10.3934/dcds.2009.25.457

[11]

Can Huang, Zhimin Zhang. The spectral collocation method for stochastic differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (3) : 667-679. doi: 10.3934/dcdsb.2013.18.667

[12]

Angelamaria Cardone, Dajana Conte, Beatrice Paternoster. Two-step collocation methods for fractional differential equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2709-2725. doi: 10.3934/dcdsb.2018088

[13]

Fuke Wu, Peter E. Kloeden. Mean-square random attractors of stochastic delay differential equations with random delay. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1715-1734. doi: 10.3934/dcdsb.2013.18.1715

[14]

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 - A, 2013, 33 (2) : 885-903. doi: 10.3934/dcds.2013.33.885

[15]

Jinyan Fan, Jianyu Pan. On the convergence rate of the inexact Levenberg-Marquardt method. Journal of Industrial & Management Optimization, 2011, 7 (1) : 199-210. doi: 10.3934/jimo.2011.7.199

[16]

Yves Bourgault, Damien Broizat, Pierre-Emmanuel Jabin. Convergence rate for the method of moments with linear closure relations. Kinetic & Related Models, 2015, 8 (1) : 1-27. doi: 10.3934/krm.2015.8.1

[17]

Tomás Caraballo, José Real, T. Taniguchi. The exponential stability of neutral stochastic delay partial differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 295-313. doi: 10.3934/dcds.2007.18.295

[18]

Min Zhu, Panpan Ren, Junping Li. Exponential stability of solutions for retarded stochastic differential equations without dissipativity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2923-2938. doi: 10.3934/dcdsb.2017157

[19]

Steven D. Galbraith, Ping Wang, Fangguo Zhang. Computing elliptic curve discrete logarithms with improved baby-step giant-step algorithm. Advances in Mathematics of Communications, 2017, 11 (3) : 453-469. doi: 10.3934/amc.2017038

[20]

Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2018, 13 (5) : 1-17. doi: 10.3934/jimo.2018037

2017 Impact Factor: 0.972

Article outline

Figures and Tables

[Back to Top]