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Weak antithetic MLMC estimation of SDEs with the Milstein scheme for low-dimensional Wiener processes. (English) Zbl 1524.60121

Summary: In this paper, we implement a weak Milstein Scheme to simulate low-dimensional stochastic differential equations (SDEs). We prove that combining the antithetic multilevel Monte-Carlo (MLMC) estimator introduced by Giles and Szpruch with the MLMC approach for weak SDE approximation methods by Belomestny and Nagapetyan, we can achieve a quadratic computational complexity in the inverse of the Root Mean Square Error (RMSE) when estimating expected values of smooth functionals of SDE solutions, without simulating Lévy areas and without requiring any strong convergence of the underlying SDE approximation method. By using appropriate discrete variables this approach allows us to calculate the expectation on the coarsest level of resolution by enumeration, which, for low-dimensional problems, results in a reduced computational effort compared to standard MLMC sampling. These theoretical results are also confirmed by a numerical experiment.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
91G20 Derivative securities (option pricing, hedging, etc.)
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References:

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[4] Giles, M. B.; Debrabant, K.; Rößler, A., Analysis of multilevel Monte Carlo using the Milstein discretisation, Discrete Contin. Dyn. Syst. Ser. B (2018), in press
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