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Efficient and Practical Implementations of Cubature on Wiener Space

In: Stochastic Analysis 2010

Author

Listed:
  • Lajos Gergely Gyurkó

    (University of Oxford, Mathematical Institute
    University of Oxford, Oxford-Man Institute)

  • Terry J. Lyons

    (University of Oxford, Mathematical Institute
    University of Oxford, Oxford-Man Institute)

Abstract

This paper explores and implements high-order numerical schemes for integrating linear parabolic partial differential equations with piece-wise smooth boundary data. The high-order Monte-Carlo methods we present give extremely accurate approximations in computation times that we believe are comparable with much less accurate finite difference and basic Monte-Carlo schemes. A key step in these algorithms seems to be that the order of the approximation is tuned to the accuracy one requires. A considerable improvement in efficiency can be attained by using ultra high-order cubature formulae. Lyons and Victoir (“Cubature on Wiener Space, Proc. R. Soc. Lond. A 460, 169–198”) give a degree 5 approximation of Brownian motion. We extend this cubature to degrees 9 and 11 in 1-dimensional space-time. The benefits are immediately apparent.

Suggested Citation

  • Lajos Gergely Gyurkó & Terry J. Lyons, 2011. "Efficient and Practical Implementations of Cubature on Wiener Space," Springer Books, in: Dan Crisan (ed.), Stochastic Analysis 2010, pages 73-111, Springer.
    • Handle: RePEc:spr:sprchp:978-3-642-15358-7_5
    DOI: 10.1007/978-3-642-15358-7_5
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