Project

Back to overview

Rare Events & Extremes of Multi-Valued Random Fields

English title Rare Events & Extremes of Multi-Valued Random Fields
Applicant Hashorva Enkelejd
Number 175752
Funding scheme Project funding
Research institution HEC - Ecole des Hautes Etudes Commerciales Université de Lausanne
Institution of higher education University of Lausanne - LA
Main discipline Mathematics
Start/End 01.01.2018 - 31.12.2020
Approved amount 193'719.00
Show all

Keywords (18)

Spectral tail process; Heavy-tailed time series; Sojourn times ; Weighted Kolmogorov test; Ruin probability; Rare events and extremes ; Max-stable random fields; Multi-valued random fields; Queueing theory; Risk theory; Excursion sets; Ruin time approximation; Extremal index ; Cummulutive Parisian ruin; Uniform double-sum method; Fractonal Brownian motion on spheres; Berman type constants; Gaussian random fields

Lay Summary (German)

Lead
Rare Ereignisse und Extremewerte multidimensionaler Zufallsfelder
Lay summary
Wichtige Klimadaten wie  Lufttemperatur, Niederschlagsniveaus und Windgeschwindigkeiten gemessen an verschiedenen Stationen sowie Daten von Beobachtungen aus verschiedenen Galaxien sind typische Beispiele von multidimensionalen Daten in Zeit und Raum.  Seltene und extreme Ereignisse sind entscheidend für das Verständnis von Trends und zugrunde liegenden Phänomenen, welche anhand den obigen genannten Daten zur Analyse stehen. Die traditionelle Analyse von raren Ereignissen und Extremen beschäftigt sich mit der Modellierung von Frequenzen seltener Ereignisse und die Severitäten der Extreme. In den mathematischen Modellen, die in diesem Projekt untersucht werden sollen, wird die Gesamtzeit der seltenen Ereignisse, die durch Extremwerte (oder hohe Schwellenwerte) verursacht werden, als die "Sojourn time" bezeichnet. Die bisherige Literatur verfügt nur über Ad-hoc Methoden, welche teilweise geignet sind, "Sojourn time" oder "occupation time" zu analysiren. Das Hauptziel dieses Projektes ist die Entwicklung der theoretischen Methodik für die probabilistische Analyse der Sojourn-time und measure of excurssion sets.
Direct link to Lay Summary Last update: 03.10.2017

Responsible applicant and co-applicants

Employees

Project partner

Publications

Publication
Multivariate extremes over a random number of observations
Hashorva Enkelejd, Padoan Simone A., Rizzelli Stefano (2021), Multivariate extremes over a random number of observations, in Scandinavian Journal of Statistics, 48(3), 845-880.
Bounds for expected supremum of fractional Brownian motion with drift
Bisewski Krzysztof, Dębicki Krzysztof, Mandjes Michel (2021), Bounds for expected supremum of fractional Brownian motion with drift, in Journal of Applied Probability, 58(2), 411-427.
Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics
Hashorva Enkelejd, Mishura Yuliya, Shevchenko Georgiy (2021), Boundary Non-crossing Probabilities of Gaussian Processes: Sharp Bounds and Asymptotics, in Journal of Theoretical Probability, 34, 728-754.
Sojourn Times of Gaussian Processes with Trend
Dȩbicki Krzysztof, Liu Peng, Michna Zbigniew (2020), Sojourn Times of Gaussian Processes with Trend, in Journal of Theoretical Probability, 33(4), 2119-2166.
Approximation of Kolmogorov–Smirnov test statistic
Bai Long, Kalaj David (2020), Approximation of Kolmogorov–Smirnov test statistic, in Stochastics, 1-35.
Simultaneous ruin probability for two-dimensional brownian risk model
Dȩbicki Krzysztof, Hashorva Enkelejd, Michna Zbigniew (2020), Simultaneous ruin probability for two-dimensional brownian risk model, in Journal of Applied Probability, 57(2), 597-612.
Asymptotic domination of sample maxima
Hashorva Enkelejd, Rullière Didier (2020), Asymptotic domination of sample maxima, in Statistics & Probability Letters, 160, 108703-108703.
Extremes of standard multifractional Brownian motion
Bai Long (2020), Extremes of standard multifractional Brownian motion, in Statistics & Probability Letters, 159, ??- ??.
Extremes of vector-valued Gaussian processes
Dȩbicki Krzysztof, Hashorva Enkelejd, Wang Longmin (2020), Extremes of vector-valued Gaussian processes, in Stochastic Processes and their Applications, 130, 5802-5837.
Tail asymptotics for Shepp-statistics of Brownian motion in $\mathbb {R}^{d}$
Korshunov Dmitry, Wang Longmin (2020), Tail asymptotics for Shepp-statistics of Brownian motion in $\mathbb {R}^{d}$, in Extremes, 23(1), 35-54.
Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants
Dȩbicki Krzysztof, Hashorva Enkelejd (2020), Approximation of Supremum of Max-Stable Stationary Processes & Pickands Constants, in Journal of Theoretical Probability, 33(1), 444-464.
Approximation of ruin probability and ruin time in discrete Brownian risk models
Jasnovidov Grigori (2020), Approximation of ruin probability and ruin time in discrete Brownian risk models, in Scandinavian Actuarial Journal, 1-18.
Estimation of change-point models
BaiLong (2020), Estimation of change-point models, in Fundamental and Applied Mathematics, 23(1), 51-73.
On the maximum of a Gaussian process with unique maximum point of its variance
HashorvaEnkelejd, KolbekovS.G., PiterbargV.I., RadinovI.V. (2020), On the maximum of a Gaussian process with unique maximum point of its variance, in Fundamental and Applied Mathematics, 23(1), 161-174.
Tail asymptotic behavior of the supremum of a class of chi-square processes
Ji Lanpeng, Liu Peng, Robert Stephan (2019), Tail asymptotic behavior of the supremum of a class of chi-square processes, in Statistics & Probability Letters, 154, 108551-108551.
Drawdown and Drawup for Fractional Brownian Motion with Trend
Bai Long, Liu Peng (2019), Drawdown and Drawup for Fractional Brownian Motion with Trend, in Journal of Theoretical Probability, 32(3), 1581-1612.
The time of ultimate recovery in Gaussian risk model
Debicki Krzysztof, Liu Peng (2019), The time of ultimate recovery in Gaussian risk model, in Extremes, 22(3), 499-521.
Extremes of spherical fractional Brownian motion
Cheng Dan, Liu Peng (2019), Extremes of spherical fractional Brownian motion, in Extremes, 22(3), 433-457.
Extremes of Gaussian chaos processes with trend
Bai Long (2019), Extremes of Gaussian chaos processes with trend, in Journal of Mathematical Analysis and Applications, 473(2), 1358-1376.
Approximation of some multivariate risk measures for Gaussian risks
Hashorva Enkelejd (2019), Approximation of some multivariate risk measures for Gaussian risks, in Journal of Multivariate Analysis, 169, 330-340.
Tail measure and spectral tail process of regularly varying time series
Dombry Clément, Hashorva Enkelejd, Soulier Philippe (2018), Tail measure and spectral tail process of regularly varying time series, in The Annals of Applied Probability, 28(6), 3884-3921.
Extremal behavior of hitting a cone by correlated Brownian motion with drift
Dȩbicki Krzysztof, Hashorva Enkelejd, Ji Lanpeng, Rolski Tomasz (2018), Extremal behavior of hitting a cone by correlated Brownian motion with drift, in Stochastic Processes and their Applications, 128, 4171-4206.
Extremes of nonstationary Gaussian fluid queues
Dȩbicki Krzysztof, Liu Peng (2018), Extremes of nonstationary Gaussian fluid queues, in Advances in Applied Probability, 50(3), 887-917.
Extremes of vector-valued Gaussian processes with Trend
Bai Long, Dȩbicki Krzysztof, Liu Peng (2018), Extremes of vector-valued Gaussian processes with Trend, in Journal of Mathematical Analysis and Applications, 465(1), 47-74.
Representations of max-stable processes via exponential tilting
Hashorva Enkelejd (2018), Representations of max-stable processes via exponential tilting, in Stochastic Processes and their Applications, 128(9), 2952-2978.
Extremes of Lp -norm of vector-valued Gaussian processes with trend
Bai Long (2018), Extremes of Lp -norm of vector-valued Gaussian processes with trend, in Stochastics, 90(8), 1111-1144.
Domination of sample maxima and related extremal dependence measures
Hashorva Enkelejd (2018), Domination of sample maxima and related extremal dependence measures, in Dependence Modelling, 88-101.
Sample path properties of reflected Gaussian processes
Kosiński Kamil Marcin, Liu Peng (2018), Sample path properties of reflected Gaussian processes, in Latin American Journal of Probability and Mathematical Statistics, (1), 453-453.

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
68e Journée de séminaires actuariels Talk given at a conference Bivariate Brownian Risk Models 21.01.2020 Lyon, France Krystecki Konrad Adam;
EVA conference 2019 Talk given at a conference Double finite time ruin probability for correlated brownian motions 03.07.2019 Zagreb, Croatia Krystecki Konrad Adam;
Dependence modelling with applications to finance and insurance Individual talk On some new extremal dependent risk measures 13.09.2018 Aegina, Greece Hashorva Enkelejd;
9TH INTERNATIONAL WORKSHOP ON APPLIED PROBABILITY(IWAP 2018) 18-21 June 2018, Budapest, Hungary Talk given at a conference Hight Excursion Probabilities of Correlated Brownian Motions 17.06.2018 Budapest, Hungary Hashorva Enkelejd;
The 40th Conference on Stochastic Processes and their Applications – SPA 2018 Talk given at a conference Hight Excursion Probabilities of Correlated Brownian Motions 11.06.2018 Gothenburg, Sweden Hashorva Enkelejd;
Stochastic Models VI Talk given at a conference Occupation times of Gaussian risk processes and Gaussian queues 03.06.2018 Bedlewp, Poland Debicki Krzysztof; Liu Peng;
Stochastic Models VI Talk given at a conference Supremum of Max-Stable Processes & Pickands-Piterbarg Constants 03.06.2018 Bedlewo, Poland Debicki Krzysztof; Hashorva Enkelejd;
Risk 2018 Talk given at a conference Form classical to Parisian Ruin: Gaussian risk Model (plenary talk) 26.04.2018 Santender, Spain Hashorva Enkelejd;


Abstract

Climate data such as air temperature, precipitation levels, and wind speeds measured at different locations on earth are typical examples of observations in time and space that can be modelled by multi-valued random fields. Other examples are total loss amounts from different insurance portfolios, the level of overloads of different queues, observation data from different galaxies, etc. Rare and extreme events are crucial for understanding trends and underlying phenomena, say global warming, or the occurrence of natural catastrophes. Traditional analysis of rare events and extremes is concerned with the modelling of frequencies of rare events and the severities of extremes. Advanced extreme value models and applications investigate further the total occurrence time of rare events which is essential for quantifying say climate changes, or for measuring the performance of stressed multiple queues. Taking again an example from climate data, observing in Switzerland air temperatures higher than 41 degrees or less than -41 degrees is quite rare, but have been already recorded! The local impact of such extremes strongly depends on the total duration in time of those occurrences. If such temperatures are observed say for only few minutes, their impact is negligible compared to extreme air temperatures observed over months or even years. In the mathematical models to be investigated in this paper, the total time of rare events caused by extreme values (or high thresholds) is referred to as the sojourn time. Observing -41.8 degrees in La Brévine on January 12, 1987 has only a local impact; occurrences of such extreme air temperatures in the whole Switzerland has much severe global impacts. Both the time and space that give rise to exceedances of high thresholds (referred to as the excursion set) pin down essential behaviours of rare events. Probabilistic analysis of sojourn times and volumes of excursion sets can be simplified by considering high thresholds.Currently in the literature, there are no developed methodologies or ad hoc results for theprobabilistic study of sojourn times and volumes of excursion sets of non-smooth multi-valued random processes and random fields, respectively. In order to analyse the probabilistic laws underlying the extreme air temperatures and extreme wind speeds, the number of observation should be considerably large. For such cases, extreme value theory justifies the use of max-stable processes and random fields as adequate limit models for spatial and temporal extremes.The main objective of this project is the development of the theoretical methodology for the probabilistic analysis of sojourn times/ volume of excursion sets of both stationary and non-stationary multi-valued random processes/fields with trend. Our asymptotic analysis will be primarily based on the {\it uniform double-sum method} for general homogeneous functionals that will be developed in this project. Additionally, this proposal includes the investigation of tractable max-stable processes/ random fields and discusses both theoretical and applied aspects related to their representations and max-domains of attraction. Further, our study includes characterisations of spectral tail processes of stationary heavy-tailed time series that are of interest for both max-stable processes and for the investigation of various asymptotic constants related to sojourn times of multi-valued random processes. Besides, this project will be concerned with the asymptotic analysis of specific complex models of rare event. In particular we shall be concerned with the extremal behaviour of Gaussian random fields on spheres, the tail behaviour of weighted Kolmogorov-Smirinov test statistics, approximation of ruin probabilities, exit and ruin times, and the performance of fluid multi-valued Gaussian queues.
-