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Efficient and Practical Implementations of Cubature on Wiener Space

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Stochastic Analysis 2010

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.

MSC (2010): 65C05, 65C30, 65M75, 91G60

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Author information

Authors and Affiliations

  1. Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX1 3LB, UK

    Lajos Gergely Gyurkó & Terry J. Lyons

  2. Oxford-Man Institute, University of Oxford, Eagle House, Walton Well Road, Oxford, OX2 6ED, UK

    Lajos Gergely Gyurkó & Terry J. Lyons

Authors
  1. Lajos Gergely Gyurkó
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  2. Terry J. Lyons
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Corresponding author

Correspondence to Terry J. Lyons .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    Dan Crisan

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Gyurkó, L.G., Lyons, T.J. (2011). Efficient and Practical Implementations of Cubature on Wiener Space. In: Crisan, D. (eds) Stochastic Analysis 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_5

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