Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises
Journal article, 2012

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 L2 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.© Springer Science+Business Media, LLC 2012.

Milstein scheme

Galerkin method

Karhunen-Loève expansion

Zakai equation

Nonequidistant time stepping

Finite element method

Advection-diffusion PDE

Stochastic partial differential equation



A. Barth

Applied Mathematics and Optimization

0095-4616 (ISSN) 1432-0606 (eISSN)

Vol. 66 3 387-413

Subject Categories

Computational Mathematics

Probability Theory and Statistics



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