Abstract
In this work, we propose a smart idea to couple importance sampling and Multilevel Monte Carlo (MLMC). We advocate a per level approach with as many importance sampling parameters as the number of levels, which enables us to handle the different levels independently. The search for parameters is carried out using sample average approximation, which basically consists in applying deterministic optimisation techniques to a Monte Carlo approximation rather than resorting to stochastic approximation. Our innovative estimator leads to a robust and efficient procedure reducing both the discretization error (the bias) and the variance for a given computational effort. In the setting of discretized diffusions, we prove that our estimator satisfies a strong law of large numbers and a central limit theorem with optimal limiting variance, in the sense that this is the variance achieved by the best importance sampling measure (among the class of changes we consider), which is however non tractable. Finally, we illustrate the efficiency of our method on several numerical challenges coming from quantitative finance and show that it outperforms the standard MLMC estimator.
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References
Alaya MB, Hajji K, Kebaier A (2015) Importance sampling and statistical Romberg method. Bernoulli 21(4):1947–1983. ISSN 1350-7265. doi:10.3150/14-BEJ622
Alaya MB, Hajji K, Kebaier A (2016) Improved adaptive Multilevel Monte Carlo and applications to finance. http://adsabs.harvard.edu/abs/2016ar**v160302959B
Alaya MB, Kebaier A (2014) Multilevel Monte Carlo for Asian options and limit theorems. Monte Carlo Methods Appl 20(3):181–194. ISSN 0929-9629. doi:10.1515/mcma-2013-0025
Alaya MB, Kebaier A (2015) Central limit theorem for the multilevel monte carlo euler method. Ann Appl Probab 2(1):211–234
Arouna B (2004) Adaptative Monte Carlo method, a variance reduction technique. Monte Carlo Methods Appl 10(1):1–24. ISSN 0929-9629. doi:10.1163/156939604323091180
Badouraly Kassim L, Lelong J, Loumrhari I (2014) Importance sampling for jump processes and applications to finance. J Comput Finance (to appear), 00(00). http://hal.archives-ouvertes.fr/hal-00842362
Chen HF, Lei G, Gao AJ (1988) Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process Appl 27(2):217–231. ISSN 0304-4149. doi:10.1016/0304-4149(87)90039-1
Chen HF, Zhu YM (1986) Stochastic approximation procedures with randomly varying truncations. Sci Sinica Ser A 29(9):914–926. ISSN 0253-5831
Collier N, Haji-Ali A-L, Nobile F, von Schwerin E, Tempone R (2015) A continuation multilevel Monte Carlo algorithm. BIT 55(2):399–432. ISSN 0006-3835. doi:10.1007/s10543-014-0511-3
Creutzig J, Dereich S, Müller-Gronbach T, Ritter K (2009) Infinite-dimensional quadrature and approximation of distributions. Found Comput Math 9(4):391–429. ISSN 1615-3375. doi:10.1007/s10208-008-9029-x
Dereich S (2011) Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann Appl Probab 21 (1):283–311. ISSN 1050-5164. doi:10.1214/10-AAP695
Duffie D, Glynn P (1995) Efficient Monte Carlo simulation of security prices. Ann Appl Probab 5(4):897–905
Giles MB (2008a) Multilevel Monte Carlo path simulation. Oper Res 56(3):607–617. ISSN 0030-364X. doi:10.1287/opre.1070.0496
Giles MB (2008b) Improved multilevel Monte Carlo convergence using the Milstein scheme Monte Carlo and quasi-Monte Carlo methods 2006. doi:10.1007/978-3-540-74496-2_20. Springer, Berlin, pp 343–358
Giles MB, Higham DJ, Mao X (2009) Analysing multi-level Monte Carlo for options with non-globally Lipschitz payoff. Finance Stoch 13(3):403–413. ISSN 0949-2984. doi:10.1007/s00780-009-0092-1
Giles MB, Szpruch L (2014) Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without Lévy area simulation. Ann Appl Probab 24 (4):1585–1620. ISSN 1050-5164. doi:10.1214/13-AAP957
Hajji K (2014) Accélération de la méthode de Monte Carlo pour des processus de diffusions et applications en Finance. PhD thesis, Université, Paris 13
Heinrich S (1998) Monte Carlo complexity of global solution of integral equations. J Complexity 14(2):151–175. ISSN 0885-064X. doi:10.1006/jcom.1998.0471
Heinrich S (2001) Multilevel monte carlo methods. Lect Notes Comput Sci, Springer-Verlag 2179(1):58–67
Heinrich S, Sindambiwe E (1999) Monte carlo complexity of parametric integration. J Complexity 15(3):317–341. ISSN 0885-064X. doi:10.1006/jcom.1999.0508. Dagstuhl Seminar on Algorithms and Complexity for Continuous Problems (1998)
Jacod J (1997) On continuous conditional Gaussian martingales and stable convergence in law Séminaire de probabilités, XXXI, volume 1655 of Lecture Notes in Math. Springer, Berlin, pp 232–246
Jacod J, Protter P (1998) Asymptotic error distributions for the Euler method for stochastic differential equations. Ann Probab 26(1):267–307. ISSN 0091-1798
Jourdain B, Lelong J (2009) Robust adaptive importance sampling for normal random vectors. Ann Appl Probab 19(5):1687–1718. doi:10.1214/09-AAP595. http://arxiv.org/pdf/0811.1496v2
Kebaier A (2005) Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing. Ann Appl Probab 15(4):2681–2705. doi:10.1214/105051605000000511
Lapeyre B, Lelong J (2011) A framework for adaptive Monte Carlo procedures. Monte Carlo Methods Appl 17(1):77–98. ISSN 0929-9629. doi:10.1515/MCMA.2011.002
Lelong J (2008) Almost sure convergence for randomly truncated stochastic algorithms under verifiable conditions. Statist Probab Lett 78(16):2632–2636. ISSN 0167-7152. doi:10.1016/j.spl.2008.02.034
Lemaire V, Pagès G (2016) Multilevel Richardson-Romberg extrapolation. Bernoulli. (to appear). 1401.1177v3
Revuz D (1997) Probabilités. Hermann
Rubinstein RY, Shapiro A (1993) Discrete event systems. Wiley series in probability and mathematical statistics: probability and mathematicalx statistics. Wiley, Chichester. Sensitivity analysis and stochastic optimization by the score function method
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We are grateful to the anonymous referees for their valuable comments and suggestions, which helped us greatly improve the paper.
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This research benefited from the support of the chair Risques Financiers, Fondation du Risque and the Laboratory of Excellence MME-DII (http://labex-mme-dii.u-cergy.fr/).
This project was supported by the Finance for Energy Market Research Centre, www.fime-lab.org.
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Kebaier, A., Lelong, J. Coupling Importance Sampling and Multilevel Monte Carlo using Sample Average Approximation. Methodol Comput Appl Probab 20, 611–641 (2018). https://doi.org/10.1007/s11009-017-9579-y
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DOI: https://doi.org/10.1007/s11009-017-9579-y
Keywords
- Sample average approximation
- Multilevel Monte Carlo
- Variance reduction
- Uniform strong large law of numbers
- Central limit theorem
- Importance sampling