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Rare-event simulation with Markov chain Monte Carlo
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematical Statistics.
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

Stochastic simulation is a popular method for computing probabilities or expecta- tions where analytical answers are difficult to derive. It is well known that standard methods of simulation are inefficient for computing rare-event probabilities and there- fore more advanced methods are needed to those problems.

This thesis presents a new method based on Markov chain Monte Carlo (MCMC) algorithm to effectively compute the probability of a rare event. The conditional distri- bution of the underlying process given that the rare event occurs has the probability of the rare event as its normalising constant. Using the MCMC methodology a Markov chain is simulated, with that conditional distribution as its invariant distribution, and information about the normalising constant is extracted from its trajectory.

In the first two papers of the thesis, the algorithm is described in full generality and applied to four problems of computing rare-event probability in the context of heavy- tailed distributions. The assumption of heavy-tails allows us to propose distributions which approximate the conditional distribution conditioned on the rare event. The first problem considers a random walk Y1 + · · · + Yn exceeding a high threshold, where the increments Y are independent and identically distributed and heavy-tailed. The second problem is an extension of the first one to a heavy-tailed random sum Y1+···+YN exceeding a high threshold,where the number of increments N is random and independent of Y1 , Y2 , . . .. The third problem considers the solution Xm to a stochastic recurrence equation, Xm = AmXm−1 + Bm, exceeding a high threshold, where the innovations B are independent and identically distributed and heavy-tailed and the multipliers A satisfy a moment condition. The fourth problem is closely related to the third and considers the ruin probability for an insurance company with risky investments.

In last two papers of this thesis, the algorithm is extended to the context of light- tailed distributions and applied to four problems. The light-tail assumption ensures the existence of a large deviation principle or Laplace principle, which in turn allows us to propose distributions which approximate the conditional distribution conditioned on the rare event. The first problem considers a random walk Y1 + · · · + Yn exceeding a high threshold, where the increments Y are independent and identically distributed and light-tailed. The second problem considers a discrete-time Markov chains and the computation of general expectation, of its sample path, related to rare-events. The third problem extends the the discrete-time setting to Markov chains in continuous- time. The fourth problem is closely related to the third and considers a birth-and-death process with spatial intensities and the computation of first passage probabilities.

An unbiased estimator of the reciprocal probability for each corresponding prob- lem is constructed with efficient rare-event properties. The algorithms are illustrated numerically and compared to existing importance sampling algorithms.

##### Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2015. , p. xii, 109
##### Series
TRITA-MAT-A ; 14:15
##### National Category
Probability Theory and Statistics
##### Research subject
Applied and Computational Mathematics
##### Identifiers
ISBN: 978-91-7595-404-2 (print)OAI: oai:DiVA.org:kth-157522DiVA, id: diva2:770640
##### Public defence
2015-01-23, F3, Lindstedtsvägen 26, KTH, Stockholm, 14:00 (English)
##### Note

QC 20141216

Available from: 2014-12-16 Created: 2014-12-10 Last updated: 2014-12-16Bibliographically approved
##### List of papers
1. Markov chain monte carlo for computing rare-event probabilities for a heavy-tailed random walk
Open this publication in new window or tab >>Markov chain monte carlo for computing rare-event probabilities for a heavy-tailed random walk
2014 (English)In: Journal of Applied Probability, ISSN 0021-9002, E-ISSN 1475-6072, Vol. 51, no 2, p. 359-376Article in journal (Refereed) Published
##### Abstract [en]

In this paper a method based on a Markov chain Monte Carlo (MCMC) algorithm is proposed to compute the probability of a rare event. The conditional distribution of the underlying process given that the rare event occurs has the probability of the rare event as its normalizing constant. Using the MCMC methodology, a Markov chain is simulated, with the aforementioned conditional distribution as its invariant distribution, and information about the normalizing constant is extracted from its trajectory. The algorithm is described in full generality and applied to the problem of computing the probability that a heavy-tailed random walk exceeds a high threshold. An unbiased estimator of the reciprocal probability is constructed whose normalized variance vanishes asymptotically. The algorithm is extended to random sums and its performance is illustrated numerically and compared to existing importance sampling algorithms.

##### Place, publisher, year, edition, pages
Applied Probability Trust, 2014
##### Keyword
Markov chain Monte Carlo, heavy tail, rare-event simulation, random walk
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:kth:diva-136800 (URN)10.1239/jap/1402578630 (DOI)000338269000005 ()2-s2.0-84904006393 (Scopus ID)
##### Note

QC 20140806

Research funded by Göran Gustafsson's foundation

Available from: 2013-12-09 Created: 2013-12-09 Last updated: 2017-12-06Bibliographically approved
2. Markov chain Monte Carlo for rare-event simulation for stochastic recurrence equations with heavy-tailed innovations
Open this publication in new window or tab >>Markov chain Monte Carlo for rare-event simulation for stochastic recurrence equations with heavy-tailed innovations
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:kth:diva-157804 (URN)
##### Note

QS 2014

Available from: 2014-12-16 Created: 2014-12-16 Last updated: 2014-12-16Bibliographically approved
3. Markov chain Monte Carlo for rare-event simulation for light-tailed random walk
Open this publication in new window or tab >>Markov chain Monte Carlo for rare-event simulation for light-tailed random walk
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:kth:diva-157805 (URN)
##### Note

QS 2014

Available from: 2014-12-16 Created: 2014-12-16 Last updated: 2014-12-16Bibliographically approved
4. Markov chain Monte Carlo for rare-event simulation for Markov chains
Open this publication in new window or tab >>Markov chain Monte Carlo for rare-event simulation for Markov chains
##### National Category
Probability Theory and Statistics
##### Identifiers
urn:nbn:se:kth:diva-157806 (URN)
##### Note

QS 2014

Available from: 2014-12-16 Created: 2014-12-16 Last updated: 2014-12-16Bibliographically approved

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Cite
Citation style
• apa
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• ieee
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Language
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