# American Institute of Mathematical Sciences

August  2019, 24(8): 3881-3903. doi: 10.3934/dcdsb.2018335

## Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation

 1 Mathematical Institute, Oxford University, Oxford OX2 6GG, United Kingdom 2 Department of Mathematics and Computer Science, University of Southern Denmark, 5230 Odense, Denmark 3 Institute of Mathematics, University of Lübeck, 23562 Lübeck, Germany

* Corresponding author: Michael B. Giles

Received  March 2018 Published  August 2019 Early access  January 2019

The multilevel Monte Carlo path simulation method introduced by Giles (Operations Research, 56(3):607-617, 2008) exploits strong convergence properties to improve the computational complexity by combining simulations with different levels of resolution. In this paper we analyse its efficiency when using the Milstein discretisation; this has an improved order of strong convergence compared to the standard Euler-Maruyama method, and it is proved that this leads to an improved order of convergence of the variance of the multilevel estimator. Numerical results are also given for basket options to illustrate the relevance of the analysis.

Citation: Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335
##### References:

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##### References:
Asian option
Lookback option
Barrier option
Digital option
Orders of convergence for $V_\ell$ as observed numerically and proved analytically for both the Euler-Maruyama and Milstein discretisations; $\delta$ can be any strictly positive constant
 Euler-Maruyama Milstein option numerical analysis numerical analysis Lipschitz ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2)$ Asian ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2)$ lookback ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2 (\log h)^2)$ barrier ${O} (h^{1/2})$ ${o} (h^{1/2-\delta})$ ${O} (h^{3/2})$ ${o} (h^{3/2-\delta})$ digital ${O} (h^{1/2})$ ${O} (h^{1/2}\log h)$ ${O} (h^{3/2})$ ${o} (h^{3/2-\delta})$
 Euler-Maruyama Milstein option numerical analysis numerical analysis Lipschitz ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2)$ Asian ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2)$ lookback ${O} (h)$ ${O} (h)$ ${O} (h^2)$ ${O} (h^2 (\log h)^2)$ barrier ${O} (h^{1/2})$ ${o} (h^{1/2-\delta})$ ${O} (h^{3/2})$ ${o} (h^{3/2-\delta})$ digital ${O} (h^{1/2})$ ${O} (h^{1/2}\log h)$ ${O} (h^{3/2})$ ${o} (h^{3/2-\delta})$
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