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. © 2011 Springer-Verlag Berlin Heidelberg.
CITATION STYLE
Gyurkó, L. G., & Lyons, T. J. (2011). Efficient and practical implementations of cubature on wiener space. In Stochastic Analysis 2010 (pp. 73–111). Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-642-15358-7_5
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