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

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:123

Article

Barth, Andrea^1,2  Lang, Annika^1,3 

[1] ETH, Seminar Angew Math, CH-8092 Zurich, Switzerland

[2] Univ Oslo, Ctr Math Applicat, N-0316 Oslo, Norway

[3] Univ Mannheim, Fak Math & Informat, D-68131 Mannheim, Germany

关键词: Stochastic partial differential equation;    L-P convergence;    Almost sure convergence;    Milstein scheme;    Galerkin method;    Finite Element method;    Backward Euler scheme;    Advection-diffusion equation;   

DOI  :  10.1016/j.spa.2013.01.003

来源: Elsevier

【 摘 要 】

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 proven. The (semidiscrete) approximation in space is a projection onto a finite dimensional function space. The considered space approximation 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-Loeve expansion of the driving noise according to the overall order of convergence. Convergence results in L-P and almost sure convergence bounds for the semidiscrete approximation as well as for the fully discrete approximation are provided. (c) 2013 Elsevier B.V. All rights reserved.

【 授权许可】

Free