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Multilevel Monte Carlo method with applications to stochastic partial differential equations

by A. Barth and A. Lang

(Report number 2011-68)

Abstract
In this work the approximation of Hilbert-space-valued random variables is combined with the approximation of the expectation by a multilevel Monte Carlo method. The number of samples on the different levels of the multilevel approximation are chosen such that the errors are balanced. The overall work then decreases in the optimal case to $O(h-2)$ if h is the error of the approximation. The multilevel Monte Carlo method is applied to functions of parabolic and hyperbolic stochastic partial differential equations as needed e.g. for option pricing. Simulations complete the paper.

Keywords: Multilevel Monte Carlo, stochastic partial differential equations, stochastic Finite Element methods, stochastic parabolic equation, stochastic hyperbolic equation, multilevel approximations.

@Techreport{BL11_124,
  author = {A. Barth and A. Lang},
  title = {Multilevel Monte Carlo method with applications to stochastic partial differential equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2011-68},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2011/2011-68.pdf },
  year = {2011}
}

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