Abstract
In this paper, we present a novel approach towards variance reduction for discretised diffusion processes. The proposed approach involves specially constructed control variates and allows for a significant reduction in the variance for the terminal functionals. In this way, the complexity order of the standard Monte Carlo algorithm (\(\varepsilon ^{-3}\)) can be reduced down to \(\varepsilon ^{-2}\sqrt{\left| \log (\varepsilon )\right| }\) in case of the Euler scheme with \(\varepsilon \) being the precision to be achieved. These theoretical results are illustrated by several numerical examples.
The study has been funded by the Russian Academic Excellence Project ‘5–100’.
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Notes
- 1.
Performing the full complexity analysis via Lagrange multipliers one can see that these parameter values are not optimal if \(2(p+1)\le d\) or \(\nu \le \frac{2d(p+1)}{2(p+1)-d}\) (a Lagrange multiplier corresponding to a ‘\(\le 0\)’ constraint is negative, cf. proof of Theorem 8). Therefore, the recommendation is to choose the power p for our basis functions according to \(p>\frac{d-2}{2}\). The opposite choice is allowed as well (the method converges), but theoretical complexity of the method would be then worse than that of the SMC, namely, \(\varepsilon ^{-3}\).
- 2.
For the Euler scheme, there is an additional logarithmic factor in the complexity of the MLMC algorithm (see [3]).
- 3.
Compare with footnote 1 on p. 16.
- 4.
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Acknowledgements
Stefan Häfner thanks the Faculty of Mathematics of the University of Duisburg-Essen, where this work was carried out.
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Belomestny, D., Häfner, S., Urusov, M. (2017). Regression-Based Variance Reduction Approach for Strong Approximation Schemes. In: Panov, V. (eds) Modern Problems of Stochastic Analysis and Statistics. MPSAS 2016. Springer Proceedings in Mathematics & Statistics, vol 208. Springer, Cham. https://doi.org/10.1007/978-3-319-65313-6_7
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