Multi-Step Maruyama Methods for Stochastic Delay Differential Equations

Abstract

In this article the numerical approximation of solutions of Itô stochastic delay differential equations is considered. We construct stochastic linear multi-step Maruyama methods and develop the fundamental numerical analysis concerning their 𝕃 _p -consistency, numerical 𝕃 _p -stability and 𝕃 _p -convergence. For the special case of two-step Maruyama schemes we derive conditions guaranteeing their mean-square consistency.

Keywords:

• Linear multi-step Maruyama methods
• Stochastic delay differential equations
• Strong convergence
• Itô formula for stochastic delay differential equations

Mathematics Subject Classification:

• 60H35
• 65C30
• 34K50
• 65L06
• 65Q05

Acknowledgments

E. Buckwar acknowledges support by DFG grant 234499. R. Winkler received support from the DFG Research Center, Mathematics for Key Technologies in Berlin. The authors thank an anonymous referee for his detailed and helpful comments.