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Multilevel path simulation for weak approximation schemes

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  • Denis Belomestny
  • Tigran Nagapetyan

Abstract

In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) methods for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler scheme for L\'evy driven stochastic differential equations, and show that, given a weak convergence of order $\alpha\geq 1/2,$ the complexity of the corresponding "weak" MLMC estimate is of order $\varepsilon^{-2}\log ^{2}(\varepsilon).$ The numerical performance of the new "weak" MLMC method is illustrated by several numerical examples.

Suggested Citation

  • Denis Belomestny & Tigran Nagapetyan, 2014. "Multilevel path simulation for weak approximation schemes," Papers 1406.2581, arXiv.org, revised Oct 2014.
    • Handle: RePEc:arx:papers:1406.2581
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    References listed on IDEAS

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    6. Michael B. Giles, 2008. "Multilevel Monte Carlo Path Simulation," Operations Research, INFORMS, vol. 56(3), pages 607-617, June.
    7. Mariko Ninomiya & Syoiti Ninomiya, 2009. "A new higher-order weak approximation scheme for stochastic differential equations and the Runge–Kutta method," Finance and Stochastics, Springer, vol. 13(3), pages 415-443, September.
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