An Exponential Wagner--Platen Type Scheme for SPDEs

Abstract

The strong numerical approximation of semilinear stochastic partial differential equations (SPDEs) driven by infinite dimensional Wiener processes is investigated. There are a number of results in the literature that show that Euler-type approximation methods converge, under suitable assumptions, to the exact solutions of such SPDEs with strong order $ {1}/{2} $ or at least with strong order $ {1}/{2} - \varepsilon$, where $ \varepsilon > 0 $ is arbitrarily small. Recent results extend these results and show that Milstein-type approximation methods converge, under suitable assumptions, to the exact solutions of such SPDEs with strong order $ 1 - \varepsilon $. It has also been shown that splitting-up approximation methods converge, under suitable assumptions, with strong order $ 1 $ to the exact solutions of such SPDEs. In this article an exponential Wagner--Platen type numerical approximation method for such SPDEs is proposed and shown to converge, under suitable assumptions, with strong order $ {3}/{2} - \varepsilon $ to the exact solutions of such SPDEs.