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2024 | OriginalPaper | Chapter

MLMC Techniques for Discontinuous Functions

Author : Michael B. Giles

Published in: Monte Carlo and Quasi-Monte Carlo Methods

Publisher: Springer International Publishing

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Abstract

The Multilevel Monte Carlo (MLMC) approach usually works well when estimating the expected value of a quantity which is a Lipschitz function of intermediate quantities, but if it is a discontinuous function it can lead to a much slower decay in the variance of the MLMC correction. This article reviews the literature on techniques which can be used to overcome this challenge in a variety of different contexts, and discusses recent developments using either a branching diffusion or adaptive sampling.

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previous chapter Quasi Continuous Level Monte Carlo for Random Elliptic PDEs
next chapter Introduction to Gaussian Process Regression in Bayesian Inverse Problems, with New Results on Experimental Design for Weighted Error Measures
Footnotes
2
Note that if f is smooth, or at least Lipschitz, then it is better to use an “antithetic” estimator [8, 14, 15, 18], but this does not give a better order of convergence when f is discontinuous.
 
Literature
1.
Achtsis, N., Cools, R., Nuyens, D.: Conditional sampling for barrier option pricing under the LT method. SIAM J. Financ. Math. 4, 327–352 (2013)MathSciNetCrossRef
2.
Altmayer, M., Neuenkirch, A.: Multilevel Monte Carlo quadrature of discontinuous payoffs in the generalized Heston model using Malliavin integration by parts. SIAM J. Financ. Math. 6(1), 22–52 (2015)MathSciNetCrossRef
3.
Asmussen, S., Glynn, P.W.: Stochastic Simulation. Springer, New York (2007)
4.
5.
6.
Bayer, C., Siebenmorgen, M., Tempone, R.: Smoothing the payoff for efficient computation of basket option prices. Quant. Financ. 18(3), 491–505 (2018)MathSciNetCrossRef
7.
Broadie, M., Du, Y., Moallemi, C.C.: Efficient risk estimation via nested sequential simulation. Manag. Sci. 57(6), 1172–1194 (2011)CrossRef
8.
Bujok, K., Hambly, B., Reisinger, C.: Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives. Methodol. Comput. Appl. Probab. 17(3), 579–604 (2015)MathSciNetCrossRef
9.
10.
Burgos, S., Giles, M.B.: Computing Greeks using multilevel path simulation. In: Plaskota, L., Woźniakowski, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 281–296. Springer (2012)
11.
Elfverson, D., Hellman, F., Målqvist, A.: A multilevel Monte Carlo method for computing failure probabilities. SIAM/ASA J. Uncertain. Quantif. 4(1), 312–330 (2016)MathSciNetCrossRef
12.
Giles, M.B.: Improved multilevel Monte Carlo convergence using the Milstein scheme. In: Keller, A., Heinrich, S., Niederreiter, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2006, pp. 343–358. Springer (2008)
13.
14.
15.
Giles, M.B.: MLMC for nested expectations. In: Contemporary Computational Mathematics-A Celebration of the 80th Birthday of Ian Sloan. Springer (2018)
16.
Giles, M.B., Bernal, F.: Multilevel estimation of expected exit times and other functionals of stopped diffusions. SIAM/ASA J. Uncertain. Quantif. 6(4), 1454–1474 (2018)MathSciNetCrossRef
17.
Giles, M.B., Debrabant, K., Rößler, A.: Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discret. Contin. Dyn. Syst. Ser. B 24(8), 3881–3903 (2019)MathSciNet
18.
Giles, M.B., Goda, T.: Decision-making under uncertainty: using MLMC for efficient estimation of EVPPI. Stat. Comput. 29(4), 739–751 (2019)MathSciNetCrossRef
19.
Giles, M.B., Haji-Ali, A.-L.: Multilevel nested simulation for efficient risk estimation. SIAM/ASA J. Uncertain. Quantif. 7(2), 497–525 (2019)MathSciNetCrossRef
20.
21.
Giles, M.B., Haji-Ali, A.-L., Spence, J.: Efficient risk estimation for the credit valuation adjustment (2023). arXiv:​2301.​05886. Accepted for publication in Annals of Applied Probability
22.
Giles, M.B., Higham, D.J., Mao, X.: Analysing multilevel Monte Carlo for options with non-globally Lipschitz payoff. Financ. Stoch. 13(3), 403–413 (2009)CrossRef
23.
Giles, M.B., Nagapetyan, T., Ritter, K.: Multilevel Monte Carlo approximation of distribution functions and densities. SIAM/ASA J. Uncertain. Quantif. 3(1), 267–295 (2015)MathSciNetCrossRef
24.
Giles, M.B., Szpruch, L.: Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann. Appl. Probab. 24(4), 1585–1620 (2014)MathSciNetCrossRef
25.
Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2004)
26.
Glasserman, P., Merener, N.: Convergence of a discretization scheme for jump-diffusion processes with state-dependent intensities. Proc. R. Soc. Lond. A 460, 111–127 (2004)MathSciNetCrossRef
27.
Haji-Ali, A.-L., Spence, J., Teckentrup, A.: Adaptive multilevel Monte Carlo for probabilities. SIAM J. Numer. Anal. 60(4), 2125–2149 (2022)MathSciNetCrossRef
28.
Kloeden, P.E., Platen, E.: Numerical Solution of Stochastic Differential Equations. Springer, Berlin (1992)CrossRef
29.
Krumscheid, S., Nobile, F.: Multilevel Monte Carlo approximation of functions. SIAM/ASA J. Uncertain. Quantif. 6(3), 1256–1293 (2018)MathSciNetCrossRef
30.
L’Ecuyer, P.: A unified view of the IPA, SF and LR gradient estimation techniques. Manag. Sci. 36(11), 1364–1383 (1990)CrossRef
31.
L’Ecuyer, P.: On the interchange of derivative and expectation for likelihood ratio derivative estimators. Manag. Sci. 41(4), 738–748 (1995)CrossRef
32.
Rhee, C.-H., Glynn, P.W.: Unbiased estimation with square root convergence for SDE models. Oper. Res. 63(5), 1026–1043 (2015)MathSciNetCrossRef
33.
34.
Xia, Y., Giles, M.B.: Multilevel path simulation for jump-diffusion SDEs. In: Plaskota, L., Woźniakowski, H. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2010, pp. 695–708. Springer (2012)
Metadata
Title
MLMC Techniques for Discontinuous Functions
Author
Michael B. Giles
Copyright Year
2024
Book
Monte Carlo and Quasi-Monte Carlo Methods

Print ISBN: 978-3-031-59761-9

Electronic ISBN: 978-3-031-59762-6

Copyright Year: 2024

DOI
https://doi.org/10.1007/978-3-031-59762-6_2

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