Modelling extremal events

Lecturer:

Václav Kůs

Tutor:

Václav Kůs

Supervisor:

Department of Mathematics

Synopsis:

The course is devoted to extremal events models. Thus, events which occur with low probability, but with significant influence on behaving of described model. Risk theory, fluctuation of random sums, and fluctuation of maxima will be taught. Further distribution for modeling extremal events and various models will be introduced. Theoretical results will be applied on real data.

Requirements:

Basic course of Calculus, Linear Algebra and Probability and Statistics

Equations (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2, 01PRST held at the FNSPE

CTU in Prague).

Syllabus of lectures:

Aim of this course is to present mathematical models for extremal events, which occur with fairly low probability, but cause a significant influence on behaving of the modeled system. And further apply them on actual problems as real data of floods, fire, financial and assurance risks,? to predict extremal events. 1. Risk Theory: classical risk models, risk process, renewal counting processes N(t) and their properties, traditional Cramér-Lundberg estimate of tail probability, time discreet sequences. 2. Fluctuation of sums: Random walk, the law of large numbers, Poisson distribution and process as limit law of counting rare events, Hartman- Winter law of iterated logarithm, functional CLT and its softening, stable and α-stable distribution and process as limit of summing process of random variables with heavy tails, spectral representation of stable distribution. 3.Fluctuation of random maxima: Gumbel, Fréchet and Weibull distribution as limit distribution of maximal value iid variables, Cramer and Heyde law of large deviations, limit laws for maxima Mn=max(X1, X2,...,Xn), Fisher-Tippett law, Poissons approximations for P(Mn .lt. un), MDA - range of stable weak convergence of maxima Mn , application on distribution and expected value of exceeding given bound. 4. Distribution for modelling extremal values: Heavy tailed probability distribution, generalized Pareto, loggama, lognormal, heavy tailed Weibull, generalized Gumbel distribution of extremal values, parameters estimates of theses distribution and their asymptotic properties, QQ plot for filtration of true distribution of extremal values. 5. Other models and their application: Model with subexponential distribution S for heavy tailed distribution, function class Rα with regular variance of the order α in infinity, Karamat theorem. 6. application on data of floods, assurance (cumulative number of insured accident) and financial risk.

Syllabus of tutorials:

Study Objective:

Distribution for modeling extremal events, properties of mentioned models, risk theory, fluctuation of random sums, and fluctuation of random maxima.

Skills: Application of given methods and models on real data with the aim to predict.

Study materials:

Key references:

[1] P. Embrechts, C. Klüppelberg, T. Mikosch Modelling Extremal Events, New York Springer 1997

Recommended references:

[2] S. Coles, An Introduction to Statistical Modeling of Extreme Values Springer-Verlag London 2001.

[3] P. Embrechts, H. Schmidli , Modelling of extremal events in insurance and finance , New York, Springer 1994.

Note:

Time-table for winter semester 2011/2012:

Time-table is not available yet

Time-table for summer semester 2011/2012:

Time-table is not available yet

The course is a part of the following study plans: