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Extremes of Threshold-Dependent Random Fields

English title Extremes of Threshold-Dependent Random Fields
Applicant Hashorva Enkelejd
Number 166274
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 01.06.2016 - 31.12.2017
Approved amount 174'300.00
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Keywords (13)

Max-stable random fields; Chaos of random fields; Ruin time approximation; Levy processes; Gaussian random fields; Exit probabilties; Rare events and extremes ; Storage process; Ruin probabilities; Threshold-dependent random field; Pickands type constants; Queueing theory; Risk theory

Lay Summary (German)

Lead
Extremewert Analyse der schwellenabhängigen Zufallsfelder
Lay summary
Klassische Wahrscheinlichkeitsmodelle der Risikotheorie sind hauptsächlich mit der Analyse der Ruinwahrscheinlichkeiten und die Zeit des Ruins  beschäftigt. Ähnliche Modelle kommen  in Finanzmathematik, Statistik sowie in der Queueing Theorie vor. Aufgrund der großen Komplexität dieser Modelle wird in der Forschung sowie bei der Anwendungen mit aproximativen Ergebnissen gearbeitet. So werden zum Beispiel Ruinwahrscheinlichkeiten aproximativ berechnet, indem man das Anfangskapital beliebig gross annimmt. Solche Ergebnisse sind oft als Benchmark sehr nützlich. Die bisherige Forschung hat ergeben, dass fortgeschrittene Modelle der Risikotheore  und Warteschlangentheorie kann durch die Einführung von schwellenabhängigen Zufallsfelder im Rahmen der asymptotischen Theorie analysiert werden. Die aktuelle Literatur bietet einige Ad-hoc-Techniken und Werkzeuge für die Untersuchung von extremen Schwellenabhängigen Zufallsfelder. Daher ist das Hauptziel dieses Projekts weiter die asymptotische Theorie der Extreme von schwellenabhängigen Zufallsfelder zu entwickeln. 

                    
Direct link to Lay Summary Last update: 26.04.2016

Responsible applicant and co-applicants

Employees

Name Institute

Publications

Publication
The Joint Distribution of Running Maximum of a Slepian Process
Deng Pingjin (2018), The Joint Distribution of Running Maximum of a Slepian Process, in Methodology and Computing in Applied Probability, 20(4), 1123-1135.
Extremes of threshold-dependent Gaussian processes
Bai Long, Dȩbicki Krzysztof, Hashorva Enkelejd, Ji Lanpeng (2018), Extremes of threshold-dependent Gaussian processes, in Science China Mathematics, 61(11), 1971-2002.
Representations of max-stable processes via exponential tilting
Hashorva E.nkelejd (2018), Representations of max-stable processes via exponential tilting, in Stochastic Processes Applications, 128(9), 2952-2978.
Asymptotics of Parisian ruin of Brownian motion risk model over an infinite-time horizon
Bai L. (2018), Asymptotics of Parisian ruin of Brownian motion risk model over an infinite-time horizon, in Scandinavian Actuarial Journal, (6), 514-528.
Some Mathematical Aspects of Price Optimisation
Hashorva E., Ratomovirija G., Tamraz M., Bai Y. (2018), Some Mathematical Aspects of Price Optimisation, in Scandinavian Actuarial Journal, (5), 379-403.
On Generalised Piterbarg Constants
Bai L., Debicki K., Hashorva E., Luo L. (2018), On Generalised Piterbarg Constants, in Methodology and Computing in Applied Probability, 20(1), 137-164.
Большие выбросы процессов гауссовского хаоса. Аппроксимация в дискретном времени
А. И. Жданов, В. И. Питербар (2018), Большие выбросы процессов гауссовского хаоса. Аппроксимация в дискретном времени, in Теория вероятн. и ее примен, 63(1), 3-28.
Approximation of Maximum of Gaussian Random Fields
Hashorva E., Seleznjev O, Tan Z. (2018), Approximation of Maximum of Gaussian Random Fields, in Journal of Mathematical Analysis Applications, 457(1), 841-867.
Extremes of Randomly Scaled Gumbel Risks
Debicki K, Farkas J., Hashorva E. (2018), Extremes of Randomly Scaled Gumbel Risks, in Journal of Mathematical Analysis Applications, 458(1), 30-42.
Comparison Inequalities for Order Statistics of Gaussian Arrays
Debicki K., Hashorva E., Ji L., Ling C. (2017), Comparison Inequalities for Order Statistics of Gaussian Arrays, in ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS, 14(1), 93-116.
Asymptotic Behavior of Reliability Function for Multidimensional Aggregated Weibull Type Reliability Indices: Analytical and Computational Methods in Probability Theory
Farkas Julia, Hashorva Enkelejd, Piterbarg Vladimir I. (2017), Asymptotic Behavior of Reliability Function for Multidimensional Aggregated Weibull Type Reliability Indices: Analytical and Computational Methods in Probability Theory, Springer International Publishing, Cham.
Extremes of gamma-reflected Gaussian processes with stationary increments
Debicki K, Hashorva E., Liu P. (2017), Extremes of gamma-reflected Gaussian processes with stationary increments, in ESAIM: Probability and Statistics, 21, 495-535.
On extremal index of max-stable processes
Debicki K., Hashorva E. (2017), On extremal index of max-stable processes, in Probability and Mathematical Statistics, 37(2), 299-317.
Tail Asymptotics of Light -tailed Weibull-like Sums
Asmussen S., Hashorva E., Laub P, Taimre T (2017), Tail Asymptotics of Light -tailed Weibull-like Sums, in Probability and Mathematical Statistics, 37(2), 235-256.
Uniform Tail Approximation of homogenous functionals of Gaussian fields
Debicki K., Hashorva E., Liu P. (2017), Uniform Tail Approximation of homogenous functionals of Gaussian fields, in Advances Applied Probability, 49(4), 1037-1066.
On some new dependence models derived from multivariate collective models in insurance applications
Hashorva E, Ratovomirija G., Tamraz M. (2017), On some new dependence models derived from multivariate collective models in insurance applications, in Scandinavian Actuarial Journal, 2017(8), 730-750.
Extremes of Gaussian random fields with regularly varying dependence structure
Debicki K., Hashorva E., Liu P. (2017), Extremes of Gaussian random fields with regularly varying dependence structure, in Extremes, 20, 333-392.
The boundary non-crossing probabilities for Slepian process
Deng PingJin (2017), The boundary non-crossing probabilities for Slepian process, in Stat. Probab. Letters, 122, 26-35.
A note on ruin problems in perturbed classical risk models
Liu Peng, Zhang Chunsheng, Ji Lanpeng (2017), A note on ruin problems in perturbed classical risk models, in Statistics & Probability Letters, 120, 28-33.
Extremes of locally stationary chi-square processes with trend
Liu P., Ji L. (2017), Extremes of locally stationary chi-square processes with trend, in Stoch Proc. Applications, 127, 497-525.
Extremes of α(t)-locally stationary Gaussian processes with non-constant variances
Bai Long (2017), Extremes of α(t)-locally stationary Gaussian processes with non-constant variances, in Journal mathematical analysis and applications, 446, 248-263.
Finite time Parisian ruin of an integrated Gaussian risk model
Peng X, Luo Li (2017), Finite time Parisian ruin of an integrated Gaussian risk model, in Statistics & Probability Letters, 124, 22-29.
Generalized Pickands constants and stationary max-stable processes
Debicki K, Engelke S., Hashorva E. (2017), Generalized Pickands constants and stationary max-stable processes, in Extremes, 20, 493-517.
Lévy-driven GPS queues with heavy-tailed input
Debicki K, Liu P (2017), Lévy-driven GPS queues with heavy-tailed input, in Queueing Systems, 85, 249-267.
Parisian ruin of the brownian motion risk model with constant force of interest
Bai Long, Luo Li (2017), Parisian ruin of the brownian motion risk model with constant force of interest, in Statistics and Probability Letters, 120, 34-44.
Extremes and limit theorems for difference of chi-type processes
Albin J.M.P., Hashorva E., Ji L, Ling C (2016), Extremes and limit theorems for difference of chi-type processes, in ESAIM Statistics & Probability, 20, 349-366.

Collaboration

Group / person Country
Types of collaboration
Lanpeng Ji (UNIL) Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel
Krzysztof Debicki/University of Wroclaw Poland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
Forschungsseminar Mathematische Statistik Individual talk From classical to parisian ruin in Gaussian risk models 14.11.2017 Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, Erdgeschoss, Erhard-Schmidt-Hörsaal, Germany Hashorva Enkelejd;
10th Extreme Value Conference Talk given at a conference Extremes of transient Gaussian fluid queues 26.06.2017 Delft TU, Netherlands Liu Peng;
10th Extreme Value Conference Talk given at a conference On Pickands and Piterbarg Constants 26.06.2017 Deltt TU, Netherlands Hashorva Enkelejd;
New Methods for Empirical Analysis of Financial Markets Individual talk Aggregation of Heavy and Light-tailed Risks 09.06.2017 Comillas, Santander, Spain Hashorva Enkelejd;
Statistical Seminar Individual talk Representation of Max-stable Processes via Exponential Tilting 13.10.2016 Bocconi University, Milano, Italy Hashorva Enkelejd;
Mathematical Department Seminar Individual talk Parisian Ruin & Parisian Ruin Time -- Approximations for Gaussian Risk Models 26.09.2016 University of Montenegro, Podgorica, Montenegro Hashorva Enkelejd;
Stochastic Models V Talk given at a conference Exact asymptotics for transient fluid model with fractional Brownian motion as input 11.09.2016 Bedlewo, Poland Liu Peng;
Stochastic Models V Individual talk On generalised Picands and Piterbarg constants 11.09.2016 Będlewo, Poland, Poland Hashorva Enkelejd;
DSA PhD Seminar, Fafleralp 2016 Talk given at a conference On the ruin probability and passage times of gamma-reflected Gaussian processes with stationary increments 24.08.2016 Hotel Fafleralp, Blatten, Switzerland Liu Peng;
3rd conference of the International Society for Non-Parametric Statistics Individual talk Ruin Probability & Ruin Time Approximation for -reflected Gaussian Risk Models with Tax 11.06.2016 Avignon, France Hashorva Enkelejd;


Abstract

Classical probabilistic models of risk theory are concerned with the analysis of ruin probabilities and the time of ruin of insurance portfolios. Similar models appear in queuing theory, financial mathematics and statistics. Due to the huge complexity of those models, current research and its applications are often concerned with asymptotic approximations of various quantities of interest. For instance, ruin probabilities are approximated and thus analysed by allowing the initial capital to grow to infinity. Previous research has shown that advanced models of risk and queueing theory can be addressed in the context of the asymptotic theory by introducing threshold-dependent random fields. Explained in the context of the approximation of ruin probabilities, this means that the imposed growth of the initial capital implies changes on the underlying risk model itself. Here the initial capital plays the role of the threshold. Indeed, threshold-dependent random processes and fields are encountered in numerous problems of mathematical statistics, finance and other research fields. The current literature offers few ad hoc techniques and tools for the study of extremes of threshold-dependent random fields, commonly assumed to be Gaussian. Therefore, the main objective of this project is to further develop the asymptotic theory of extremes of threshold-dependent random fields by considering also tractable non-Gaussian random fields. The developed theory will then be applied to several open problems of risk theory, queueing theory and mathematical statistics. We plan also to fill some gaps in the current literature concerned with extremes of processes with trend. In our findings the so-called Pickands-type constants will appear, which can be calculated by simulations. This project shall investigate in detail several Pickands-type constants by studying certain hidden connections with extreme value theory of max-stable random fields. The envisaged results are of great theoretical importance and can be used for efficient simulations of those unknown constants. In addition to numerous theoretical results and their interpretation, this project shall develop new techniques and extensions that are of interest in various applications of asymptotic theory and extreme value theory.
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