Simulacija rijetkih događaja

Vazdar, Vedrana (2016) Simulacija rijetkih događaja. Diploma thesis, Faculty of Science > Department of Mathematics.

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Abstract

The probability of exceptionally rare events is difficult to estimate accurately using the standard Monte Carlo algorithm due to a high relative error of such estimators. The aim of this master’s thesis is to present some simulation techniques which may overcome this problem, with the emphasis on the importance sampling and conditional Monte Carlo method as variance reduction techniques. Algorithm efficiency measures, such as the bounded relative error and logarithmic efficiency, are precisely defined. Several algorithms are described, mainly concerning hitting times probabilities in light and heavy tailed random walks, and their theoretical efficiency proven. For light tailed distributions, the algorithms are based on the exponential change of measure in the importance sampling. The Siegmund’s algorithm is presented, for which the optimal importance distribution has been determined. A brief explanation of large deviations approach to optimal exponential change of measure is also provided. Algorithms for heavy tailed distributions are substantially less developed and it is often difficult to find a good importance distribution. This paper presents some algorithms in the case of subexponential distributions (with the emphasis on Pareto and Weibull distribution), which are based on the conditional Monte Carlo method and importance sampling. Several examples of real life application of the algorithms are discussed, such as ruin probabilities in insurance. Finally, the algorithms have been implemented in MATLAB and compared in simulations. The Siegmund’s algorithm has also proven good in practice, whereas in heavy tailed case the conditional Monte Carlo algorithms have shown better performance than the importance sampling ones.

Item Type: Thesis (Diploma thesis)
Supervisor: Basrak, Bojan
Date: 2016
Number of Pages: 49
Subjects: NATURAL SCIENCES > Mathematics
Divisions: Faculty of Science > Department of Mathematics
Depositing User: Iva Prah
Date Deposited: 22 Nov 2016 14:35
Last Modified: 22 Nov 2016 14:35
URI: http://digre.pmf.unizg.hr/id/eprint/5331

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