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Min-max representations of viscosity solutions of Hamilton-Jacobi equations and applications in rare-event simulation
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0002-6608-0715
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-9210-121X
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.ORCID iD: 0000-0001-8702-2293
(English)Manuscript (preprint) (Other academic)
Abstract [en]

In this paper a duality relation between the Mañé potential and Mather's action functional is derived in the context of convex and state-dependent Hamiltonians. The duality relation is used to obtain min-max representations of viscosity solutions of first order Hamilton-Jacobi equations. These min-max representations naturally suggest classes of subsolutions of Hamilton-Jacobi equations that arise in the theory of large deviations. The subsolutions, in turn, are good candidates for designing efficient rare-event simulation algorithms.

Keyword [en]
Hamilton-Jacobi equations, viscosity solutions, Monte Carlo methods
National Category
Probability Theory and Statistics Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-144419OAI: oai:DiVA.org:kth-144419DiVA: diva2:713461
Note

QS 2014

Available from: 2014-04-23 Created: 2014-04-23 Last updated: 2014-04-24Bibliographically approved
In thesis
1. 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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Djehiche, BoualemHult, Henrik

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