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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 & 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.  Google Scholar

[2]

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

[3]

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

[4]

I. Gyöngy, A note on Euler's approximations, Potential Anal., 8 (1998), 205-216. doi: 10.1023/A:1008605221617.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

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

[3]

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

[4]

I. Gyöngy, A note on Euler's approximations, Potential Anal., 8 (1998), 205-216. doi: 10.1023/A:1008605221617.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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