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Moderate deviation principles for importance sampling estimators of risk measures

Part of: Applications Probabilistic methods, simulation and stochastic differential equations Parametric inference Limit theorems

Published online by Cambridge University Press:  22 June 2017

Pierre Nyquist
Pierre Nyquist*
Affiliation:
Brown University
*
* Current address: Department of Mathematics, KTH Royal Institute of Technology, 100 44, Stockholm, Sweden. Email address: pierren@kth.se
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Abstract

Importance sampling has become an important tool for the computation of extreme quantiles and tail-based risk measures. For estimation of such nonlinear functionals of the underlying distribution, the standard efficiency analysis is not necessarily applicable. In this paper we therefore study importance sampling algorithms by considering moderate deviations of the associated weighted empirical processes. Using a delta method for large deviations, combined with classical large deviation techniques, the moderate deviation principle is obtained for importance sampling estimators of two of the most common risk measures: value at risk and expected shortfall.

Keywords

Large deviationmoderate deviationrisk measureempirical processasymptoticsimportance samplingMonte Carlo

MSC classification

Primary: 60F10: Large deviations 65C05: Monte Carlo methods
Secondary: 62F12: Asymptotic properties of estimators 62P05: Applications to actuarial sciences and financial mathematics
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 2 , June 2017 , pp. 490 - 506
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Arcones, M. A. (2003). Moderate deviations of empirical processes. In Stochastic Inequalities and Applications (Progr. Prob. 56), Birkhäuser, Basel, pp. 189–212. CrossRefGoogle Scholar
[2] Blanchet, J. and Glynn, P. (2008). Efficient rare-event simulation for the maximum of a heavy-tailed random walks. Ann. Appl. Prob. 18, 1351–1378. CrossRefGoogle Scholar
[3] Blanchet, J. H. and Liu, J. (2008). State-dependent importance sampling for regularly varying random walks. Adv. Appl. Prob. 40, 1104–1128. CrossRefGoogle Scholar
[4] Blanchet, J., Glynn, P. and Leder, K. (2012). On Lyapunov inequalities and subsolutions for efficient importance sampling. ACM Trans. Model. Comput. Simul. 22, 1104–1128. CrossRefGoogle Scholar
[5] Borovkov, A. A. and Mogul'skiĭ, A. A. (1980). Probabilities of large deviations in topological spaces. II. Sibirsk. Mat. Zh. 21, 12–26, 189 (in Russian). Google Scholar
[6] De Acosta, A. (1997). Moderate deviations for empirical measures of Markov chains: lower bounds. Ann. Prob. 25, 259–284. CrossRefGoogle Scholar
[7] De Acosta, A. and Chen, X. (1998). Moderate deviations for empirical measures of Markov chains: upper bounds. J. Theoret. Prob. 11, 1075–1110. CrossRefGoogle Scholar
[8] Del Moral, P., Hu, S. and Wu, L. (2015). Moderate deviations for interacting processes. Statistica Sinica 25, 921–951. Google Scholar
[9] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications (Appl. Math. 38), 2nd edn. Springer, New York. CrossRefGoogle Scholar
[10] Douc, R., Guillin, A. and Najim, A. (2005). Moderate deviations for particle filtering. Ann. Appl. Prob. 15, 587–614. CrossRefGoogle Scholar
[11] Dudley, R. M. (1999). Uniform Central Limit Theorems (Camb. Stud. Adv. Math. 63). Cambridge University Press. CrossRefGoogle Scholar
[12] Dupuis, P. and Wang, H. (2004). Importance sampling, large deviations, and differential games. Stoch. Stoch. Report 76, 481–508. CrossRefGoogle Scholar
[13] Dupuis, P. and Wang, H. (2007). Subsolutions of an Isaacs equation and efficient schemes for importance sampling. Math. Operat. Res. 32, 723–757. CrossRefGoogle Scholar
[14] Gao, F. and Wang, S. (2011). Asymptotic behavior of the empirical conditional value-at-risk. Insurance Math. Econom. 49, 345–352. CrossRefGoogle Scholar
[15] Gao, F. and Zhao, X. (2011). Delta method in large deviations and moderate deviations for estimators. Ann. Statist. 39, 1211–1240. CrossRefGoogle Scholar
[16] Glasserman, P., Heidelberger, P. and Shahabuddin, P. (2002). Portfolio value-at-risk with heavy-tailed risk factors. Math. Finance 12, 239–269. CrossRefGoogle Scholar
[17] Glynn, P. W. (1996). Importance sampling for Monte Carlo estimation of quantiles. In Proc. 2nd Internat. Workshop Mathematical Methods in Stochastic Simulation and Experimental Design, Petersburg University Press, pp. 180–185. Google Scholar
[18] Hult, H. and Nyquist, P. (2016). Large deviations for weighted empirical measures arising in importance sampling. Stoch. Process Appl. 126, 138–170. CrossRefGoogle Scholar
[19] Hult, H. and Svensson, J. (2009). Efficient calculation of risk measures by importance sampling the heavy-tailed case. Preprint. Available at https://arxiv.org/abs/0909.3335v1. Google Scholar
[20] Ledoux, M. (1992). Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. Ann. Inst. H. Poincaré Prob. Statist. 28, 267–280. Google Scholar
[21] Ledoux, M. and Talagrand, M. (2011). Probability in Banach Spaces. Springer, Berlin. Google Scholar
[22] Resnick, S. I. (2008). Extreme Values, Regular Variation and Point Processes. Springer, New York. Google Scholar
[23] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, New York. CrossRefGoogle Scholar
[24] Wu, L. (1994). Large deviations, moderate deviations and LIL for empirical processes. Ann. Prob. 22, 17–27. CrossRefGoogle Scholar