Backward Euler-Maruyama method applied to nonlinear hybrid stochastic differential equations with time-variable delay
|
摘要
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
In this paper, we consider strong convergence and almost sure exponential stability of the backward Euler-Maruyama method for nonlinear hybrid stochastic differential equations with time-variable delay. Under the local Lipschitz condition and polynomial growth condition, it is proved that the backward Euler-Maruyama method is strongly convergent. Additionally, the moment estimates and almost sure exponential stability for the analytical solution are proved. Also, under the appropriate condition, we show that the numerical solutions for the backward Euler-Maruyama methods are almost surely exponentially stable. A numerical experiment is given to illustrate the computational effectiveness and the theoretical results of the method.
引文
1 Anderson W J.Continuous-Time Markov Chains.Berlin:Springer-Verlag,1991
2 Bahar A,Mao X.Stochastic delay Lotka-Volterra model.J Math Anal Appl,2004,292:364-380
3 Bao J,Hou Z.An analytic approximation of solutions of stochastic differential delay equations with Markovian switching.Math Comput Modelling,2009,50:1379-1384
4 Basak G K,Bisi A,Ghosh M K.Stability of a random diffusion with linear drift.J Math Anal Appl,1996,202:604-622
5 Hu L,Mao X,Zhang L.Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations.IEEE Trans Automat Control,2013,58:2319-2332
6 Li H,Xiao L,Ye J.Strong predictor-corrector Euler-Maruyama methods for stochastic differential equations with Markovian switching.J Comput Appl Math,2013,237:5-17
7 Li R,Hou Y.Convergence and stability of numerical solutions to SDDEs with Markovian switching.Appl Math Comput,2006,175:1080-1091
8 Li R,Meng H,Qin C.Exponential stability of numerical solutions to SDDEs with Markovian switching.Appl Math Comput,2006,174:1302-1313
9 Mao X.Stability of stochastic differential equations with Markovian switching.Stochastic Process Appl,1999,79:45-67
10 Mao X.Stochastic Differential Equations and Applications.Cambridge:Woodhead Publishing,2007
11 Mao X.Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions.Appl Math Comput,2011,217:5512-5524
12 Mao X,Shen Y,Gray A.Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations.J Comput Appl Math,2011,235:1213-1226
13 Mao X,Szpruch L.Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients.J Comput Appl Math,2013,238:14-28
14 Mao X,Szpruch L.Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients.Stochastics,2013,85:144-171
15 Mao X,Yuan C.Stochastic Differential Equations with Markovian Switching.London:Imperial College Press,2006
16 Miloˇsevi′c M.Highly nonlinear neutral stochastic differential equations with time-dependent delay and the EulerMaruyama method.Math Comput Modelling,2011,54:2235-2251
17 Miloˇsevi′c M.Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation.Math Comput Modelling,2013,57:887-899
18 Miloˇsevi′c M.Implicit numerical methods for highly nonlinear neutral stochastic differential equations with timedependent delay.Appl Math Comput,2014,244:741-760
19 Niu Y,Burrage K,Zhang C.Multi-scale approach for simulating time-delay biochemical reaction systems.IET Syst Biol,2015,9:31-38
20 Niu Y,Zhang C,Burrage K.Strong predictor-corrector approximation for stochastic delay differential equations.JComput Math,2015,33:587-605
21 Pang S,Deng F,Mao X.Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations.J Comput Appl Math,2008,213:127-141
22 Rodkina A,Schurz H.Almost sure asymptotic stability of drift-implicitθ-methods for bilinear ordinary stochastic differential equations in R1.J Comput Appl Math,2005,180:13-31
23 Szpruch L,Mao X,Higham D J,et al.Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model.BIT,2011,51:405-425
24 Xie Y,Zhang C.Asymptotical boundedness and moment exponential stability for stochastic neutral differential equations with time-variable delay and markovian switching.Appl Math Lett,2017,70:46-51
25 Yuan C,Glover W.Approximate solutions of stochastic differential delay equations with Markovian switching.JComput Appl Math,2006,194:207-226
26 Zhou S.Strong convergence and stability of backward Euler-Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation.Calcolo,2015,52:445-473
27 Zhou S,Hu C.Numerical approximation of stochastic differential delay equation with coefficients of polynomial growth.Calcolo,2017,54:1-22
2 Bahar A,Mao X.Stochastic delay Lotka-Volterra model.J Math Anal Appl,2004,292:364-380
3 Bao J,Hou Z.An analytic approximation of solutions of stochastic differential delay equations with Markovian switching.Math Comput Modelling,2009,50:1379-1384
4 Basak G K,Bisi A,Ghosh M K.Stability of a random diffusion with linear drift.J Math Anal Appl,1996,202:604-622
5 Hu L,Mao X,Zhang L.Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations.IEEE Trans Automat Control,2013,58:2319-2332
6 Li H,Xiao L,Ye J.Strong predictor-corrector Euler-Maruyama methods for stochastic differential equations with Markovian switching.J Comput Appl Math,2013,237:5-17
7 Li R,Hou Y.Convergence and stability of numerical solutions to SDDEs with Markovian switching.Appl Math Comput,2006,175:1080-1091
8 Li R,Meng H,Qin C.Exponential stability of numerical solutions to SDDEs with Markovian switching.Appl Math Comput,2006,174:1302-1313
9 Mao X.Stability of stochastic differential equations with Markovian switching.Stochastic Process Appl,1999,79:45-67
10 Mao X.Stochastic Differential Equations and Applications.Cambridge:Woodhead Publishing,2007
11 Mao X.Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions.Appl Math Comput,2011,217:5512-5524
12 Mao X,Shen Y,Gray A.Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations.J Comput Appl Math,2011,235:1213-1226
13 Mao X,Szpruch L.Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients.J Comput Appl Math,2013,238:14-28
14 Mao X,Szpruch L.Strong convergence rates for backward Euler-Maruyama method for non-linear dissipative-type stochastic differential equations with super-linear diffusion coefficients.Stochastics,2013,85:144-171
15 Mao X,Yuan C.Stochastic Differential Equations with Markovian Switching.London:Imperial College Press,2006
16 Miloˇsevi′c M.Highly nonlinear neutral stochastic differential equations with time-dependent delay and the EulerMaruyama method.Math Comput Modelling,2011,54:2235-2251
17 Miloˇsevi′c M.Almost sure exponential stability of solutions to highly nonlinear neutral stochastic differential equations with time-dependent delay and the Euler-Maruyama approximation.Math Comput Modelling,2013,57:887-899
18 Miloˇsevi′c M.Implicit numerical methods for highly nonlinear neutral stochastic differential equations with timedependent delay.Appl Math Comput,2014,244:741-760
19 Niu Y,Burrage K,Zhang C.Multi-scale approach for simulating time-delay biochemical reaction systems.IET Syst Biol,2015,9:31-38
20 Niu Y,Zhang C,Burrage K.Strong predictor-corrector approximation for stochastic delay differential equations.JComput Math,2015,33:587-605
21 Pang S,Deng F,Mao X.Almost sure and moment exponential stability of Euler-Maruyama discretizations for hybrid stochastic differential equations.J Comput Appl Math,2008,213:127-141
22 Rodkina A,Schurz H.Almost sure asymptotic stability of drift-implicitθ-methods for bilinear ordinary stochastic differential equations in R1.J Comput Appl Math,2005,180:13-31
23 Szpruch L,Mao X,Higham D J,et al.Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model.BIT,2011,51:405-425
24 Xie Y,Zhang C.Asymptotical boundedness and moment exponential stability for stochastic neutral differential equations with time-variable delay and markovian switching.Appl Math Lett,2017,70:46-51
25 Yuan C,Glover W.Approximate solutions of stochastic differential delay equations with Markovian switching.JComput Appl Math,2006,194:207-226
26 Zhou S.Strong convergence and stability of backward Euler-Maruyama scheme for highly nonlinear hybrid stochastic differential delay equation.Calcolo,2015,52:445-473
27 Zhou S,Hu C.Numerical approximation of stochastic differential delay equation with coefficients of polynomial growth.Calcolo,2017,54:1-22