Strong Convergence of Euler Approximations of Stochastic Differential Equations with Delay Under Local Lipschitz Condition

Abstract

The strong convergence of Euler approximations of stochastic delay differential equations is proved under general conditions. The assumptions on drift and diffusion coefficients have been relaxed to include polynomial growth and only continuity in the arguments corresponding to delays. Furthermore, the rate of convergence is obtained under one-sided and polynomial Lipschitz conditions. Finally, our findings are demonstrated with the help of numerical simulations.

Keywords:

• Euler approximations
• Local Lipschitz condition
• Rate of convergence
• Stochastic delay differential equations
• Strong convergence

Mathematics Subject Classification:

• Primary 60H35
• Secondary 65C30, 34K50

Acknowledgment

The authors would like to thank Lukas Szpruch for his useful discussions.

Notes

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lsaa