Deterministic multi-level algorithms for infinite-dimensional integration on RN

  • Authors:

  • Ben Niu;Fred J. Hickernell;Thomas Müller-Gronbach;Klaus Ritter

  • Affiliations:

  • Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA;Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA;Fakultät für Informatik und Mathematik, Universität Passau, 94030, Passau, Germany;Fachbereich Mathematik, Technische Universität Kaiserslautern, Postfach 3049, 67653 Kaiserslautern, Germany

  • Venue:
  • Journal of Complexity
  • Year:
  • 2011

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Abstract

Pricing a path-dependent financial derivative, such as an Asian option, requires the computation of E(g(B)), the expectation of a payoff function g, that depends on a Brownian motion B. Employing a standard series expansion of B the latter problem is equivalent to the computation of the expectation of a function of the corresponding i.i.d. sequence of random coefficients. This motivates the construction and the analysis of algorithms for numerical integration with respect to a product probability measure on the sequence space R^N. The class of integrands studied in this paper is the unit ball in a reproducing kernel Hilbert space obtained by superposition of weighted tensor product spaces of functions of finitely many variables. Combining tractability results for high-dimensional integration with the multi-level technique we obtain new algorithms for infinite-dimensional integration. These deterministic multi-level algorithms use variable subspace sampling and they are superior to any deterministic algorithm based on fixed subspace sampling with respect to the respective worst case error.