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Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises

by A. Barth and A. Lang

(Report number 2011-36)

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 adapted 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. Finally, simulations complete the paper.

Keywords: Finite Element method, stochastic partial differential equation, martingale, Galerkin method, Zakai equation, advection-diffusion PDE, Milstein scheme, Crank--Nicolson approximation, Karhunen-Loève expansion, adapted time stepping

@Techreport{BL11_134,
  author = {A. Barth and A. Lang},
  title = {Milstein approximation for advection-diffusion equations driven by multiplicative noncontinuous martingale noises},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-36},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-36.pdf },
  year = {2011}
}

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