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The Pathwise Convergence of Approximation Schemes for Stochastic Differential Equations

2007; London Mathematical Society; Volume: 10; Linguagem: Inglês

10.1112/s1461157000001388

1461-1570

Peter E. Kloeden, Andreas Neuenkirch,

Advanced Mathematical Modeling in Engineering

Abstract The authors of this paper study approximation methods for stochastic differential equations, and point out a simple relation between the order of convergence in the p th mean and the order of convergence in the pathwise sense: Convergence in the p th mean of order α for all p ≥ 1 implies pathwise convergence of order α – ε for arbitrary ε > 0. The authors then apply this result to several one-step and multi-step approximation schemes for stochastic differential equations and stochastic delay differential equations. In addition, they give some numerical examples.