Efficient and Practical Implementations of Cubature on Wiener Space

Lajos Gergely Gyurkó () and Terry J. Lyons ()
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Lajos Gergely Gyurkó: University of Oxford, Mathematical Institute
Terry J. Lyons: University of Oxford, Mathematical Institute

A chapter in Stochastic Analysis 2010, 2011, pp 73-111 from Springer

Abstract: 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.

Keywords: Stochastic differential equation; Numerical solution; Weak approximation; Cubature; Wiener space; Expected signature; High order (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-15358-7_5

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DOI: 10.1007/978-3-642-15358-7_5

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