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Lp and almost sure convergence of a Milstein scheme for stochastic partial differential equations

by A. Barth and A. Lang

(Report number 2011-15)

Abstract
In this paper $L^p$ convergence and almost sure convergence of the Milstein approximation of a partial differential equation of advection-diffusion type driven by a multiplicative continuous martingale is proved. The (semidiscrete) approximation in space is a projection onto a finite dimensional subspace. The space approximation considered has to have an order of convergence fitting to the order of convergence of the Milstein approximation and the regularity of the solution. The approximation of the driving noise process is realized by the truncation of the Karhunen-Loève expansion according to the overall order of convergence. Convergence in $L^p$ and almost sure convergence of the semidiscrete approximation as well as of the fully discrete approximation are provided.

Keywords: Stochastic partial differential equation, $L^p$ convergence, almost sure convergence, Milstein scheme, Galerkin method, Finite Element method, Crank-Nicolson scheme

@Techreport{BL11_143,
  author = {A. Barth and A. Lang},
  title = {Lp and almost sure convergence of a Milstein scheme for stochastic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-15.pdf },
  year = {2011}
}

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