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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

Bahareh Akhtari 1, , Esmail Babolian 1, and  Andreas Neuenkirch 2,

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.
Keywords: pathwise convergence, domain, locally Lipschitz, Stochastic delay differential equations, reflection..
Mathematics Subject Classification: 60H35, 65C30, 65C2.
Citation: Bahareh Akhtari, Esmail Babolian, Andreas Neuenkirch. An Euler scheme for stochastic delay differential equations on unbounded domains: Pathwise convergence. Discrete and 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), 35-66.

[2]

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

[3]

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

[4]

I. Gyöngy, A note on Euler's approximations, Potential Anal., 8 (1998), 205-216. 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), 391-412. 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), 1041-1063. 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), 375-404. 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), 41-64. 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), 1245-1260. 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), 235-253. doi: 10.1112/S1461157000001388.

[12]

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

[13]

X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215-227. 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), 296-320. 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), 97-118. 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), 35-66.

[2]

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

[3]

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

[4]

I. Gyöngy, A note on Euler's approximations, Potential Anal., 8 (1998), 205-216. 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), 391-412. 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), 1041-1063. 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), 375-404. 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), 41-64. 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), 1245-1260. 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), 235-253. doi: 10.1112/S1461157000001388.

[12]

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

[13]

X. Mao and S. Sabanis, Numerical solutions of stochastic differential delay equations under local Lipschitz condition, J. Comput. Appl. Math., 151 (2003), 215-227. 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), 296-320. 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), 97-118. doi: 10.1142/S0219493710002875.

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