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Applied Mathematics & Optimization

, Volume 66, Issue 3, pp 387–413 | Cite as

Milstein Approximation for Advection-Diffusion Equations Driven by Multiplicative Noncontinuous Martingale Noises

  • Andrea BarthEmail author
  • Annika Lang
Article

Abstract

In this paper, the strong approximation of a stochastic partial differential equation, whose differential operator is of advection-diffusion type and which is driven by a multiplicative, infinite dimensional, càdlàg, square integrable martingale, is presented. A finite dimensional projection of the infinite dimensional equation, for example a Galerkin projection, with nonequidistant time stepping is used. Error estimates for the discretized equation are derived in L 2 and almost sure senses. Besides space and time discretizations, noise approximations are also provided, where the Milstein double stochastic integral is approximated in such a way that the overall complexity is not increased compared to an Euler–Maruyama approximation. Finally, simulations complete the paper.

Keywords

Finite element method Stochastic partial differential equation Martingale Galerkin method Zakai equation Advection-diffusion PDE Milstein scheme Karhunen–Loève expansion Nonequidistant time stepping 

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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Seminar für Angewandte MathematikETHZürichSwitzerland

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