Article Contents
Article Contents

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

• 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.
Mathematics Subject Classification: 60H35, 65C30, 65C20.

 Citation:

•  [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.