# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - B, 2019, 24 (2) : 695-717. doi: 10.3934/dcdsb.2018203
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##### References:
(a) The mean square errors. (b) The 3th moment errors
(a) The mean square errors. (b) The 3th moment errors
(a) $a = -3,~b = 0,~c = 1$. (b) $a = -1.8,~b = 0.4,~c = 0.7$
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 $2^{-6}$ $0.2330e-04$ $*$ $0.2121e-04$ $*$ $0.1974e-04$ $*$ $2^{-7}$ $0.1037e-04$ $2.2469$ $0.0982e-04$ $2.1599$ $0.0943e-04$ $2.0933$ $2^{-8}$ $0.0444e-04$ $2.3356$ $0.0435e-04$ $2.2575$ $0.0414e-04$ $2.2778$ $2^{-9}$ $0.0184e-04$ $2.4130$ $0.0179e-04$ $2.4302$ $0.0175e-04$ $2.3657$ $2^{-10}$ $0.0096e-04$ $1.9167$ $0.0096e-04$ $1.8646$ $0.0096e-04$ $1.8229$ $2^{-11}$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$ $0.0040e-04$ $2.4000$
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 $2^{-6}$ $0.4083e-06$ $*$ $0.4032e-06$ $*$ $0.4039e-06$ $*$ $2^{-7}$ $0.1266e-06$ $3.2251$ $0.1245e-06$ $3.2386$ $0.1203e-06$ $3.3574$ $2^{-8}$ $0.0373e-06$ $3.3941$ $0.0364e-06$ $3.4203$ $0.0352e-06$ $3.4176$ $2^{-9}$ $0.0127e-06$ $2.9370$ $0.0115e-06$ $3.1652$ $0.0093e-06$ $3.7849$ $2^{-10}$ $0.0067e-06$ $1.8955$ $0.0055e-06$ $2.0909$ $0.0049e-06$ $1.8980$ $2^{-11}$ $0.0011e-06$ $6.0909$ $0.0011e-06$ $5.0000$ $0.0011e-06$ $4.4545$
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 $2^{-6}$ $1.0839e-04$ $*$ $1.0689e-03$ $*$ $1.0578e-03$ $*$ $2^{-7}$ $5.1071e-04$ $2.1223$ $5.0147e-04$ $2.1315$ $4.9992e-04$ $2.1159$ $2^{-8}$ $2.6099e-04$ $1.9568$ $2.5515e-04$ $2.1000$ $2.5056e-04$ $1.9952$ $2^{-9}$ $1.2395e-04$ $2.1056$ $1.2150e-04$ $1.9158$ $1.1603e-04$ $2.1592$ $2^{-10}$ $0.6654e-04$ $1.8628$ $0.6342e-04$ $1.8646$ $0.5864e-04$ $1.9787$ $2^{-11}$ $0.3179e-04$ $2.0931$ $0.3155e-04$ $2.0101$ $0.3124e-04$ $1.8770$
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 $2^{-6}$ $3.4673e-05$ $*$ $3.4000e-05$ $*$ $3.3598e-05$ $*$ $2^{-7}$ $1.0647e-05$ $3.2566$ $1.0435e-05$ $3.2583$ $1.0150e-05$ $3.3101$ $2^{-8}$ $3.3059e-06$ $3.2206$ $3.2294e-06$ $3.2313$ $3.0933e-06$ $3.2813$ $2^{-9}$ $1.0265e-06$ $3.2206$ $1.0258e-06$ $3.1481$ $0.9983e-06$ $3.0986$ $2^{-10}$ $0.3531e-06$ $2.9071$ $0.3518e-06$ $2.9159$ $0.3426e-06$ $2.9139$ $2^{-11}$ $0.1560e-06$ $2.2635$ $0.1556e-06$ $2.2609$ $0.1515e-06$ $2.2614$
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