January  2015, 20(1): 23-38. doi: 10.3934/dcdsb.2015.20.23

An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence

1. 

Department of Mathematical Sciences and Computer, Kharazmi University, 50 Taleghani Avenue, Tehran 1561836314, Iran, Iran

2. 

Institut für Mathematik, Universität Mannheim, A5, 6, D-63181 Mannheim, Germany

Received  June 2013 Revised  June 2014 Published  November 2014

In this note, we establish under mild smoothness assumptions the pathwise convergence rate of an Euler-type method with projection for delay stochastic differential equations on unbounded domains.
Citation: Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 23-38. doi: 10.3934/dcdsb.2015.20.23
References:
[1]

J. A. D. Appleby and A. Rodkina, Asymptotic stability of polynomial stochastic delay differential equations with damped perturbations,, Funct. Differ. Equ., 12 (2005), 35.

[2]

L. Arnold, Random Dynamical Systems,, Springer, (1998). doi: 10.1007/978-3-662-12878-7.

[3]

V. I. Bogachev, Measure Theory. Vol I,, Springer, (2007). doi: 10.1007/978-3-540-34514-5.

[4]

I. Gyöngy, A note on Euler's approximations,, Potential Anal., 8 (1998), 205. doi: 10.1023/A:1008605221617.

[5]

I. Gyöngy and S. Sabanis, A note on Euler approximations for stochastic differential equations with delay,, Appl. Math. Optim., 68 (2013), 391. doi: 10.1007/s00245-013-9211-7.

[6]

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

[7]

M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, to appear in Mem. Amer. Math. Soc., ().

[8]

A. Jentzen, Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients,, Potential. Anal., 31 (2009), 375. doi: 10.1007/s11118-009-9139-3.

[9]

A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients,, Numer. Math., 112 (2009), 41. doi: 10.1007/s00211-008-0200-8.

[10]

P. E. Kloeden, G. Lord, A. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds,, J. Comput. Appl. Math., 235 (2011), 1245. doi: 10.1016/j.cam.2010.08.011.

[11]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations,, LMS J. Comput. Math., 10 (2007), 235. doi: 10.1112/S1461157000001388.

[12]

X. Mao, Stochastic Differential Equations and their Applications,, Horwood Publishing, (1997).

[13]

X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition,, J. Comput. Appl. Math., 151 (2003), 215. doi: 10.1016/S0377-0427(02)00750-1.

[14]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics,, J. Math. Anal. Appl., 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027.

[15]

F. Wu and S. Hu, A study of a class of nonlinear stochastic delay differential equations,, Stoch. Dyn., 10 (2010), 97. doi: 10.1142/S0219493710002875.

show all references

References:
[1]

J. A. D. Appleby and A. Rodkina, Asymptotic stability of polynomial stochastic delay differential equations with damped perturbations,, Funct. Differ. Equ., 12 (2005), 35.

[2]

L. Arnold, Random Dynamical Systems,, Springer, (1998). doi: 10.1007/978-3-662-12878-7.

[3]

V. I. Bogachev, Measure Theory. Vol I,, Springer, (2007). doi: 10.1007/978-3-540-34514-5.

[4]

I. Gyöngy, A note on Euler's approximations,, Potential Anal., 8 (1998), 205. doi: 10.1023/A:1008605221617.

[5]

I. Gyöngy and S. Sabanis, A note on Euler approximations for stochastic differential equations with delay,, Appl. Math. Optim., 68 (2013), 391. doi: 10.1007/s00245-013-9211-7.

[6]

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

[7]

M. Hutzenthaler and A. Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients,, to appear in Mem. Amer. Math. Soc., ().

[8]

A. Jentzen, Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients,, Potential. Anal., 31 (2009), 375. doi: 10.1007/s11118-009-9139-3.

[9]

A. Jentzen, P. E. Kloeden and A. Neuenkirch, Pathwise approximation of stochastic differential equations on domains: higher order convergence rates without global Lipschitz coefficients,, Numer. Math., 112 (2009), 41. doi: 10.1007/s00211-008-0200-8.

[10]

P. E. Kloeden, G. Lord, A. Neuenkirch and T. Shardlow, The exponential integrator scheme for stochastic partial differential equations: pathwise error bounds,, J. Comput. Appl. Math., 235 (2011), 1245. doi: 10.1016/j.cam.2010.08.011.

[11]

P. E. Kloeden and A. Neuenkirch, The pathwise convergence of approximation schemes for stochastic differential equations,, LMS J. Comput. Math., 10 (2007), 235. doi: 10.1112/S1461157000001388.

[12]

X. Mao, Stochastic Differential Equations and their Applications,, Horwood Publishing, (1997).

[13]

X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition,, J. Comput. Appl. Math., 151 (2003), 215. doi: 10.1016/S0377-0427(02)00750-1.

[14]

X. Mao, C. Yuan and J. Zou, Stochastic differential delay equations of population dynamics,, J. Math. Anal. Appl., 304 (2005), 296. doi: 10.1016/j.jmaa.2004.09.027.

[15]

F. Wu and S. Hu, A study of a class of nonlinear stochastic delay differential equations,, Stoch. Dyn., 10 (2010), 97. doi: 10.1142/S0219493710002875.

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