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Moderate deviation principles for importance sampling estimators of risk measures
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-8702-2293
2017 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072Article in journal (Refereed) Accepted
Abstract [en]

Importance sampling has become an important tool for the computation of tail-based risk measures. Since such quantities are often determined mainly by rare events standard Monte Carlo can be inefficient and importance sampling provides a way to speed up computations. This paper considers moderate deviations for the weighted empirical process, the process analogue of the weighted empirical measure, arising in importance sampling. The moderate deviation principle is established as an extension of existing results. Using a delta method for large deviations established by Gao and Zhao (Ann. Statist., 2011) together with classical large deviation techniques, the moderate deviation principle for the weighted empirical process is extended to functionals of the weighted empirical process which correspond to risk measures. The main results are moderate deviation principles for importance sampling estimators of the quantile function of a distribution and Expected Shortfall.

Place, publisher, year, edition, pages
2017.
Keyword [en]
Large deviations, moderate deviations, empirical processes, importance sampling, risk measures
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:kth:diva-117808OAI: oai:DiVA.org:kth-117808DiVA: diva2:603119
Note

QCR 20161219

Available from: 2013-02-05 Created: 2013-02-05 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Large deviations for weighted empirical measures and processes arising in importance sampling
Open this publication in new window or tab >>Large deviations for weighted empirical measures and processes arising in importance sampling
2013 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers related to large deviation results associated with importance sampling algorithms. As the need for efficient computational methods increases, so does the need for theoretical analysis of simulation algorithms. This thesis is mainly concerned with algorithms using importance sampling. Both papers make theoretical contributions to the development of a new approach for analyzing efficiency of importance sampling algorithms by means of large deviation theory.

In the first paper of the thesis, the efficiency of an importance sampling algorithm is studied using a large deviation result for the sequence of weighted empirical measures that represent the output of the algorithm. The main result is stated in terms of the Laplace principle for the weighted empirical measure arising in importance sampling and it can be viewed as a weighted version of Sanov's theorem. This result is used to quantify the performance of an importance sampling algorithm over a collection of subsets of a given target set as well as quantile estimates. The method of proof is the weak convergence approach to large deviations developed by Dupuis and Ellis.

The second paper studies moderate deviations of the empirical process analogue of the weighted empirical measure arising in importance sampling. Using moderate deviation results for empirical processes the moderate deviation principle is proved for weighted empirical processes that arise in importance sampling. This result can be thought of as the empirical process analogue of the main result of the first paper and the proof is established using standard techniques for empirical processes and Banach space valued random variables. The moderate deviation principle for the importance sampling estimator of the tail of a distribution follows as a corollary. From this, moderate deviation results are established for importance sampling estimators of two risk measures: The quantile process and Expected Shortfall. The results are proved using a delta method for large deviations established by Gao and Zhao (2011) together with more classical results from the theory of large deviations.

The thesis begins with an informal discussion of stochastic simulation, in particular importance sampling, followed by short mathematical introductions to large deviations and importance sampling.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. iv, 20 p.
Series
Trita-MAT, ISSN 1401-2286 ; 13:01
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-117810 (URN)978-91-7501-644-3 (ISBN)
Presentation
2013-02-25, 3721, Lindstedtsvägen 25, KTH, Stockholm, 15:15 (English)
Opponent
Supervisors
Note

QC 20130205

Available from: 2013-02-05 Created: 2013-02-05 Last updated: 2013-02-05Bibliographically approved
2. On large deviations and design of efficient importance sampling algorithms
Open this publication in new window or tab >>On large deviations and design of efficient importance sampling algorithms
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of four papers, presented in Chapters 2-5, on the topics large deviations and stochastic simulation, particularly importance sampling. The four papers make theoretical contributions to the development of a new approach for analyzing efficiency of importance sampling algorithms by means of large deviation theory, and to the design of efficient algorithms using the subsolution approach developed by Dupuis and Wang (2007).

In the first two papers of the thesis, the random output of an importance sampling algorithm is viewed as a sequence of weighted empirical measures and weighted empirical processes, respectively. The main theoretical results are a Laplace principle for the weighted empirical measures (Paper 1) and a moderate deviation result for the weighted empirical processes (Paper 2). The Laplace principle for weighted empirical measures is used to propose an alternative measure of efficiency based on the associated rate function.The moderate deviation result for weighted empirical processes is an extension of what can be seen as the empirical process version of Sanov's theorem. Together with a delta method for large deviations, established by Gao and Zhao (2011), we show moderate deviation results for importance sampling estimators of the risk measures Value-at-Risk and Expected Shortfall.

The final two papers of the thesis are concerned with the design of efficient importance sampling algorithms using subsolutions of partial differential equations of Hamilton-Jacobi type (the subsolution approach).

In Paper 3 we show a min-max representation of viscosity solutions of Hamilton-Jacobi equations. In particular, the representation suggests a general approach for constructing subsolutions to equations associated with terminal value problems and exit problems. Since the design of efficient importance sampling algorithms is connected to such subsolutions, the min-max representation facilitates the construction of efficient algorithms.

In Paper 4 we consider the problem of constructing efficient importance sampling algorithms for a certain type of Markovian intensity model for credit risk. The min-max representation of Paper 3 is used to construct subsolutions to the associated Hamilton-Jacobi equation and the corresponding importance sampling algorithms are investigated both theoretically and numerically.

The thesis begins with an informal discussion of stochastic simulation, followed by brief mathematical introductions to large deviations and importance sampling. 

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. xii, 20 p.
Series
TRITA-MAT-A, 14:05
Keyword
Large deviations, Monte Carlo methods, importance sampling
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:kth:diva-144423 (URN)978-91-7595-130-0 (ISBN)
Public defence
2014-05-14, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20140424

Available from: 2014-04-24 Created: 2014-04-23 Last updated: 2014-04-24Bibliographically approved

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