Derandomization of the Euler scheme for scalar stochastic differential equations

  • Authors:

  • Thomas Müller-Gronbach;Klaus Ritter;Larisa Yaroslavtseva

  • Affiliations:

  • Fakultät für Informatik und Mathematik, Universität Passau, Innstr.33, 94032 Passau, Germany;Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany;Fakultät für Informatik und Mathematik, Universität Passau, Innstr.33, 94032 Passau, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2012

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Abstract

Consider a scalar stochastic differential equation with solution process X. We present a deterministic algorithm to approximate the marginal distribution of X at t=1 by a discrete distribution, and hereby we get a deterministic quadrature rule for expectations E(f(X(1))). The construction of the algorithm is based on derandomization of the Euler scheme. We provide a worst case analysis for the computational cost and the error, assuming that the coefficients of the equation have bounded derivatives up to order four and that the derivatives of f are polynomially bounded up to order four. In terms of the computational cost the error is almost of the order 2/3, if the diffusion coefficient is bounded away from zero, and in general we almost achieve the order 1/2.