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Efficient almost-exact Lévy area sampling

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  • Malham, Simon J.A.
  • Wiese, Anke

Abstract

We present a new method for sampling the Lévy area for a two-dimensional Wiener process conditioned on its endpoints. An efficient sampler for the Lévy area is required to implement a strong Milstein numerical scheme to approximate the solution of a stochastic differential equation driven by a two-dimensional Wiener process whose diffusion vector fields do not commute. Our method is simple and complementary to those of Gaines–Lyons and Wiktorsson, and amenable to quasi-Monte Carlo implementation. It is based on representing the Lévy area by an infinite weighted sum of independent Logistic random variables. We use Chebyshev polynomials to approximate the inverse distribution function of sums of independent Logistic random variables in three characteristic regimes. The error is controlled by the degree of the polynomials, we set the error to be uniformly 10−12. We thus establish a strong almost-exact Lévy area sampling method. The complexity of our method is square logarithmic. We indicate how it can contribute to efficient sampling in higher dimensions.

Suggested Citation

  • Malham, Simon J.A. & Wiese, Anke, 2014. "Efficient almost-exact Lévy area sampling," Statistics & Probability Letters, Elsevier, vol. 88(C), pages 50-55.
    • Handle: RePEc:eee:stapro:v:88:y:2014:i:c:p:50-55
    DOI: 10.1016/j.spl.2014.01.022
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    References listed on IDEAS

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    1. P. E. Kloeden & Eckhard Platen & I. W. Wright, 1992. "The approximation of multiple stochastic integrals," Published Paper Series 1992-2, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    2. Simon J. A. Malham & Anke Wiese, 2013. "Chi-Square Simulation Of The Cir Process And The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(03), pages 1-38.
    3. Rydén, Tobias & Wiktorsson, Magnus, 2001. "On the simulation of iterated Itô integrals," Stochastic Processes and their Applications, Elsevier, vol. 91(1), pages 151-168, January.
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    Cited by:

    1. Lay Harold A. & Colgin Zane & Reshniak Viktor & Khaliq Abdul Q. M., 2018. "On the implementation of multilevel Monte Carlo simulation of the stochastic volatility and interest rate model using multi-GPU clusters," Monte Carlo Methods and Applications, De Gruyter, vol. 24(4), pages 309-321, December.
    2. Simon J. A. Malham & Jiaqi Shen & Anke Wiese, 2020. "Series expansions and direct inversion for the Heston model," Papers 2008.08576, arXiv.org, revised Jan 2021.

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