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Extremes of Gaussian Processes and Related Random Fields

English title Extremes of Gaussian Processes and Related Random Fields
Applicant Hashorva Enkelejd
Number 140633
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 01.08.2012 - 31.10.2015
Approved amount 470'301.00
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Keywords (13)

Extreme values ; Seleznjev Theorem; Interpolation of random fields; Conditional Gaussian process; Time discretisation of random fields; Perturbed risk process; Gaussian random field; Storage process; Shepp statistics; Chi-square process; Limit theorems for Gaussian processes; Weak and strong dependence; Locally-stationary random field

Lay Summary (English)

Lead
Lay summary

The classical Central Limit Theorem and its ramifications show that the Gaussian model is a natural and correct paradigm for building an approximate solution to many otherwise unsolvable problems encountered in various research fields.

Indeed, the range of applications of Gaussian processes and related random fields encompasses almost any field of theoretical and applied research. Some  extraordinary examples include variations of Brownian motion as the unique solution to problems from theoretical physics, biology, mathematical statistics, risk theory, stochastic finance, telecommunication,  just to name a few. While the theory of Gaussian processes and random fields is well-developed and mature, the range of applications of Gaussian random fields is constantly growing. Recently, applications in brain mapping, cosmology, and quantum chaos have been added to its palmares.  Due to the presence of measurement errors, missing observations or random inflations, in some cases the Gaussian framework appears as not tenable.

This project advocates that by extending the models to vector-valued chi-processes, vector-valued conditional Gaussian processes and random fields, the Gaussian framework proves to be very reliable. Essentially, numerous applications are intrinsically connected to the study of extremes of Gaussian processes and their related random fields. A natural extreme-value problem in this context is the determination of the exact tail asymptotic behaviour of the maxima of Gaussian processes over some given sets, the hardest and oldest problem in the study of random processes. Besides the tail asymptotics of the maximum, the derivation of limit theorems regarding the maxima of Gaussian processes is both of theoretical and applied interest.

This project aims at studying extremes of such large classes of vector-valued Gaussian processes, chi-processes, and conditional Gaussian processes over continuous, discrete and random sets. The principal theoretical findings envisaged by this study shall include both exact tail asymptotic results and limit theorems for the maxima of the mentioned Gaussian and related processes. Since real data are only possible to be observed on a certain discrete grid of time-points, it is planed to investigate the joint asymptotic behaviour of maximum over continuous time intervals with maxima over discrete grids, for several classes of Gaussian processes and chi-processes. Motivated by various applications in risk theory, queueing theory, and hydrodynamics this project is also concerned with
the study of the maximum of Gaussian processes and chi-processes over random time intervals. In addition to numerous theoretical results and their interpretation, this project shall develop novel methodologies and techniques. Furthermore, the derivation of some key asymptotical results for the extremes of several Gaussian fields will open the way for novel statistical applications, whereas by focusing on both Gaussian perturbed risk processes and generalsations of the storage processes, additional applications concerned with the risk analysis, simulation of rare-events and the analysis of overflows in hydrodynamics will be promoted.

Direct link to Lay Summary Last update: 21.02.2013

Responsible applicant and co-applicants

Employees

Publications

Publication
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 and Probability Letters, 120, 28-33.
Aggregation of randomly weighted large risks
Asimit V, Hashorva E, Kortschak D (2017), Aggregation of randomly weighted large risks, in IMA J Management Math, 28(3), 403-419.
A characterization of the normal distribution using stationary max-stable processes
Engelke S, Kabluchko Z. (2016), A characterization of the normal distribution using stationary max-stable processes, in Extremes, 19(1), 1-6.
A Levy process on the real line seen from its supremum and max-stable processes
Engelke S., Ivanovs J. (2016), A Levy process on the real line seen from its supremum and max-stable processes, in Electronic J. Probabiliy, 21(paper no. ), 1-19.
Exact simulation of max-stable processes
Dombry C., Engelke S., Oesting M. (2016), Exact simulation of max-stable processes, in Biometrika, 106, 303-317.
Extremes of -locally stationary Gaussian random fields
Hashorva E., Ji L. (2016), Extremes of -locally stationary Gaussian random fields, in Transactions of American Mathematica Soc., 368(1), 1-26.
Extremes of a class of non-homogeneous Gaussian random fields
Debicki K., Hashorva E., Ji L. (2016), Extremes of a class of non-homogeneous Gaussian random fields, in Annals of Probability, 44(2), 984-1012..
Extremes of Chi-square Processes with trend
Liu Peng, Ji Lanpeng (2016), Extremes of Chi-square Processes with trend, in Probability and Mathematical Statistics, 36(1), 1-20.
Extremes of stationary Gaussian storage models
Debicki K., Liu P. (2016), Extremes of stationary Gaussian storage models, in Extremes, 19(2), 273-302.
Extremes on river networks
Asadi P, Davision A, Engelke S (2016), Extremes on river networks, in Annals of Applied Statistics, 9(4), 2023-2050.
Higher-order expansions of distributions of maxima in a Hüsler-Reiss model
Hashorva E, Peng Z, Weng Z (2016), Higher-order expansions of distributions of maxima in a Hüsler-Reiss model, in Methodology and Computing in Applied Probability, 18(1), 181-196.
Maxima of skew elliptical triangular arrays
Hashorva Enkelejd, Ling Chengxiu (2016), Maxima of skew elliptical triangular arrays, in Communications in Statistics - Theory and Methods, 45(12), 3692-3705.
On maxima of chi-processes over threshold dependent grids
Tan Z., Ling C. (2016), On maxima of chi-processes over threshold dependent grids, in Statistics: A Journal of Theoretical and Applied Statistics, 50(3), 579-595.
On Parisian ruin over a finite-time horizon
Debicki K., Hashorva E., Ji L. (2016), On Parisian ruin over a finite-time horizon, in Science China Mathematics, 59(3), 557-572.
Tail asymptotics of generalized deflated risks with insurance applications
Ling Chengxiu, Peng Zuoxiang (2016), Tail asymptotics of generalized deflated risks with insurance applications, in Insurance Mathematics & Economics, 71, 220-231.
Approximation of a random process with variable smoothness
Hashorva E., Lifshits M., Seleznjev O. (2015), Approximation of a random process with variable smoothness, in Hallin M. (ed.), Springer Verlag, Germany, 189-208.
Boundary non-crossing probabilities for fractional Brownian motion with trend
Hashorva Enkelejd, Mishura Yulyia, Seleznjev Oleg (2015), Boundary non-crossing probabilities for fractional Brownian motion with trend, in Stochastics, 87(6), 946-965.
Estimation of Hüsler-Reiss distributions and Brown-Resnick processes
Enkelke S., Malinovski A., Kabluchko Z., Schlather M. (2015), Estimation of Hüsler-Reiss distributions and Brown-Resnick processes, in Journal of the Royal Statistical Society B, 77, 239-265.
Extremal Behavior of Gaussian Chaos via Probabilistic Approach
Hashorva Enkelejd, Korshunov Dmitry, Piterbarg Vladimir I. (2015), Extremal Behavior of Gaussian Chaos via Probabilistic Approach, in Extremes, 18(3), 315-347.
Extremal behavior of squared Bessel processes attracted by the Brown-Resnick process
Das Bikram, Engelke Sebastian, Hashorva Enkelejd (2015), Extremal behavior of squared Bessel processes attracted by the Brown-Resnick process, in Stochastic Processes and their Applications, 125(2), 780-796.
Extremes of Aggregated Dirichlet Risks
Enkelejd Hashorva (2015), Extremes of Aggregated Dirichlet Risks, in J. Multivariate Analysis, 133, 334-345.
Extremes of homogeneous Gaussian random fields
Dębicki K., Hashorva E., Soja-Kukieła N. (2015), Extremes of homogeneous Gaussian random fields, in Journal of Applied Probability, 52(1), 55-67.
Extremes of order statistics of stationary processes
Debicki K., Hashorva E., Ji L., Ling C. (2015), Extremes of order statistics of stationary processes, in Test, 24(2), 229-248.
Extremes of vector-valued Gaussian processes: Exact asymptotics
Debicki K., Hashorva E., Ji L., Tabis K. (2015), Extremes of vector-valued Gaussian processes: Exact asymptotics, in Stochastic Proc Applications, 125(11), 4039-4065.
Gaussian risk models with financial constraints
Debicki K., Hashorva E., Ji L. (2015), Gaussian risk models with financial constraints, in Scandinavian Actuarial Journal, 6, 469-481.
Maxima of a triangular array of multivariate Gaussian sequence
Hashorva Enkelejd, Peng Liang, Weng Zhichao (2015), Maxima of a triangular array of multivariate Gaussian sequence, in Statistics & Probability Letters, 103, 62-72.
Max-stable processes and stationary systems of Levy particles
Engelke S., Kabluchko Z. (2015), Max-stable processes and stationary systems of Levy particles, in Stochastic Proc Applications, 125(11), 4272-4299.
On Laplace asymptotic method, with application to random chaos
Korshunov Dmitry, Piterbarg Vladimir I., Hashorva Enkelejd (2015), On Laplace asymptotic method, with application to random chaos, in Matematicheskie Zametki, 97(6), 868-883.
On Sarmanov Mixed Erlang Risks in Insurance Applications
Hashorva E., Ratovomirija G (2015), On Sarmanov Mixed Erlang Risks in Insurance Applications, in ASTIN Bulletin, 45(1), 175-205.
On the gamma-reflected processes with fBm input
Peng L., Hashorva E., Ji L. (2015), On the gamma-reflected processes with fBm input, in Lithianian Math J., 55(3), 402-412.
Parisian ruin of self-similar Gaussian risk processes
Debicki K., Hashorva E., Ji L. (2015), Parisian ruin of self-similar Gaussian risk processes, in J. Applied Probability, 53(3), 688-702.
Piterbarg theorems for chi-processes with trend
Hashorva Enkelejd, Ji Lanpeng (2015), Piterbarg theorems for chi-processes with trend, in Extremes, 37-64.
Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids
Hashorva E., Tan Z. (2015), Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids, in Statistics, 49(2), 338-360.
Tail approximation for reinsurance portfolios of Gaussian-like risks
Farkas Julia, Hashorva Enkelejd (2015), Tail approximation for reinsurance portfolios of Gaussian-like risks, in Scandinavian Actuarial Journal, 4, 319-331.
Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks
Hashorva Enkelejd, Li Jinzhu (2015), Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks, in Stochastic Models, 31(1), 1-19.
Tail dependence for two skew slash distributions
Chengxiu Ling, Zuoxiang Peng (2015), Tail dependence for two skew slash distributions, in Statistics and Its Interface, 8(1), 63-69.
Tail dependence for two skew slash distributions
Ling Chengxiu, Peng Zuoxiang (2015), Tail dependence for two skew slash distributions, in Statistics and its Interfaces, 8(1), 63-69.
Tail asymptotic expansions for L-statisitcs.
Hashorva Enkelejd, Ling Chengxiu, Peng Zuoxiang (2014), Tail asymptotic expansions for L-statisitcs., in Science China Mathematics, 57(10), 1993-2012.
A duality result for the generalized Erlang risk model
Ji Lanpeng, Zhang Chunsheng (2014), A duality result for the generalized Erlang risk model, in Risks, 2, 456-466.
Aggregation of log-linear risks
Embrechts P., Hashorva E., Mikosch T. (2014), Aggregation of log-linear risks, in J. Appl. Probab., 51(A), 203-212.
Approximation of passage times of gamma-reflected processes with fBm input
Hashorva Enkelejd, Ji Lanpeng (2014), Approximation of passage times of gamma-reflected processes with fBm input, in Journal of Applied Probability, 51(3), 713-726.
Asymptotics for a Discrete-time Risk Model with the Emphasis on Financial Risk
Hashorva E, Li J (2014), Asymptotics for a Discrete-time Risk Model with the Emphasis on Financial Risk, in Probability in the Engineering and Informational Sciences, 28(4), 573-588.
Asymptotics of the finite-time ruin probability for the Sparre Andersen risk model perturbed by an inflated stationary chi-process
Hashorva Enkelejd, Ji Lanpeng (2014), Asymptotics of the finite-time ruin probability for the Sparre Andersen risk model perturbed by an inflated stationary chi-process, in Communications in Statistics - Theory and Methods, 43, 2540-2548.
Berman's inequality under random scaling
Enkelejd Hashorva, Zhichao Weng (2014), Berman's inequality under random scaling, in Statistics and Its Interface, 7(3), 339-349.
Boundary Non-Crossings of Additive Wiener Fields
Hashorva E., Mishura Y. (2014), Boundary Non-Crossings of Additive Wiener Fields, in Lithuanian Math. J., 54(3), 277-289.
EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FRACTIONAL BROWNIAN MOTIONS
Hashorva Enkelejd, Ji Lanpeng (2014), EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FRACTIONAL BROWNIAN MOTIONS, in STOCHASTIC MODELS, 30(3), 272-299.
Extremes of perturbed bivariate Rayleigh risks
Hashorva Enkelejd, Nadarajah Saralees, Pogany K Tibor (2014), Extremes of perturbed bivariate Rayleigh risks, in REVSTAT, 12(1), 157-168.
Finite-time ruin probability of aggregate Gaussian processes
Debicki K., Hashorva E., Ji L., Tan Z. (2014), Finite-time ruin probability of aggregate Gaussian processes, in Markov Processes and Related Fields, 20, 435-450.
Gaussian approximation of perturbed chi-square risks
Debicki Krzysztof, Hashorva Enkelejd, Ji Lanpeng (2014), Gaussian approximation of perturbed chi-square risks, in Statistics and Its Interface, 7(3), 363-373.
Limit properties of exceedances point processes of scaled stationary Gaussian sequences
Hashorva Enkelejd, Peng Zuoxiang, Weng Zhichao (2014), Limit properties of exceedances point processes of scaled stationary Gaussian sequences, in Probability and Mathematical Statistics, 34(1), 45-59.
Maxima and minima of complete and incomplete stationary sequences
Hashorva Enkelejd, Weng Zhichao (2014), Maxima and minima of complete and incomplete stationary sequences, in Stochastics An International Journal of Probability and Stochastic Processes, 86(5), 707-720.
Modeling of Censored Bivariate Extremal Events
Hashorva Enkelejd, Ling Chengxiu, Peng Zuoxiang (2014), Modeling of Censored Bivariate Extremal Events, in Journal of the Korean Statistical Society, 43(3), 323-338.
On Piterbarg max-discretisation theorem for standardised maximum of stationary gaussian processes
Tan Zhongquan, Hashorva Enkelejd (2014), On Piterbarg max-discretisation theorem for standardised maximum of stationary gaussian processes, in Methodology and Computing in Applied Probability, 16(1), 169-185.
On the probability of conjunctions of stationary Gaussian processes
Debicki Krzysztof, Hashorva Enkelejd, Ji Lanpeng, Tabis Kamil (2014), On the probability of conjunctions of stationary Gaussian processes, in Statistics & Probability Letters, 88, 141-148.
Random shifting and scaling of insurance risks
Hashorva Enkelejd, Ji Lanpeng (2014), Random shifting and scaling of insurance risks, in Risks, 2, 277-288.
Second order asymptotics of aggregated log-elliptical risk
Kortschak Dominik, Hashorva Enkelejd (2014), Second order asymptotics of aggregated log-elliptical risk, in Methodology and Computing in Applied Probability, 16(4), 969-985.
Tail asymptotics of random sum and maximum of log-normal risks
Hashorva Enkelejd, Kortschak Dominik (2014), Tail asymptotics of random sum and maximum of log-normal risks, in Statist. Probab. Letters, 87, 167-174.
Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals
Dȩbicki Krzysztof, Hashorva Enkelejd, Ji Lanpeng (2014), Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals, in Extremes, 17(3), 411-429.
Large deviations of Shepp statistics for fractional Brownian motion
Hashorva Enkelejd, Tan Zhongquan (2013), Large deviations of Shepp statistics for fractional Brownian motion, in Statistics & Probability Letters, 83(10), 2242-2247.
Aggregation of parametrised log-elliptical risks
Hashorva Enkelejd (2013), Aggregation of parametrised log-elliptical risks, in Journal of Mathematical Analysis and Applications, 400(1), 187-199.
ECOMOR and LCR reinsurance with gamma-like claims
Hashorva Enkelejd, Li Jinzhu (2013), ECOMOR and LCR reinsurance with gamma-like claims, in Insurance: Mathematics and Economics, 53(1), 206-215.
Efficient simulation of tail probabilities for sums of log-elliptical risks
Kortschak Dominik, Hashorva Enkelejd (2013), Efficient simulation of tail probabilities for sums of log-elliptical risks, in Journal of Computational and Applied Mathematics, 247(1), 53-67.
Exact asymptotics and limit theorems for supremum of stationary chi-processes over a random interval.
Tan Zhongquan, Hashorva Enkelejd (2013), Exact asymptotics and limit theorems for supremum of stationary chi-processes over a random interval., in Stoch. Processes Applications, 123(8), 2983-2998.
Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval
Tan Zhongquan, Hashorva Enkelejd (2013), Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval, in Lithuanian Mathematical Journal, 53(1), 91-102.
Limit theorems for extremes of strongly dependent cyclo-stationary chi-processes
Tan Zhongquan, Hashorva Enkelejd (2013), Limit theorems for extremes of strongly dependent cyclo-stationary chi-processes, in Extremes, 16(2), 241-254.
On Extremal Behavior of Gaussian Chaos
Korshunov D.A., Piterbarg V.I., Hashorva E. (2013), On Extremal Behavior of Gaussian Chaos, in Doklady Mathematics, 88(2), 566-568.
On Piterbarg theorem for the maxima of stationary Gaussian sequences
Hashorva Enkelejd, Peng Zuoxiang, Weng Zhichao (2013), On Piterbarg theorem for the maxima of stationary Gaussian sequences, in Lithuanian Math Journal, 53(3), 280-292.
On Piterbarg's max-discretisation theorem for multivariate stationary Gaussian processes
Tan Zhongquan, Hashorva Enkelejd (2013), On Piterbarg's max-discretisation theorem for multivariate stationary Gaussian processes, in Journal of Mathematical Analysis and Applications, 409(1), 299-314.
On the supremum of gamma-reflected processes with fractional Brownian motion as input
Hashorva Enkelejd, Ji Lanpeng, Piterbarg I Vladimir (2013), On the supremum of gamma-reflected processes with fractional Brownian motion as input, in Stochastic Processes and their Applications, 123(11), 4111-4127.
Asymptotics of maxima of strongly dependent Gaussian processes
Tan Zhongquan, Hashorva Enkelejd, Peng Zuoxiang (2012), Asymptotics of maxima of strongly dependent Gaussian processes, in J. Appl. Probab., 49(4), 1106-1118.
Calculation of Bayes premium for conditionally elliptical risks
Kume Alfred, Hashorva Enkelejd (2012), Calculation of Bayes premium for conditionally elliptical risks, in Insurance: Mathematics and Economics, 51, 632-635.

Collaboration

Group / person Country
Types of collaboration
Southwest University China (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel
University Tor Vergata, Rome Italy (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Bern Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Georgia Institute of Technology, Atlanta United States of America (North America)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Moscow State University Russia (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Rostock Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results
Nankai University China (Asia)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel
Chalmers University of Technology Sweden (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Copenhagen Denmark (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Wroclaw Poland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
- Exchange of personnel
National Taras Shevchenko University of Kyiv Ukraine (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
University of Umea Sweden (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Sobolev Institute of Mathematics Russia (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
ETH Zurich Switzerland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
CAS Business School Great Britain and Northern Ireland (Europe)
- in-depth/constructive exchanges on approaches, methods or results
- Publication
Katholische Universität Eichstätt-Ingolstadt Germany (Europe)
- in-depth/constructive exchanges on approaches, methods or results

Scientific events

Active participation

Title Type of contribution Title of article or contribution Date Place Persons involved
EVA 2015 Individual talk Extremes of a class of non-homogeneous Gaussian random fields 15.06.2015 Ann Arbor, Michigan Univeristiy, United States of America Engelke Sebastian; Hashorva Enkelejd;
55e Journée de séminaires actuariels Lyon-Lausanne Individual talk Approximations of Weyl Fractional Integrals with Insurance Applications 21.01.2015 Lausanne, Switzerland Ling Chengxiu;
Statistics, Probability & Numerical Analysis Individual talk ASYMTOTIC EXPANSIONS OF RUIN PROBABILITY FOR TIME-CHANGED fBM RISK PROCESSES 05.12.2014 Faculty of Natural Sciences, University of Tirama, Tirana, Albania Hashorva Enkelejd;
Colloquia on Probability and Statistics Individual talk Max-stable Processes on River Networks 26.09.2014 Bern, Switzerland Engelke Sebastian;
Joint Statistical Meeting 2014 Talk given at a conference Max-Stable Processes on River Networks 07.08.2014 Boston, United States of America Engelke Sebastian;
Probability Seminar, Nankai University, Mathematical School Individual talk Asymptotic Expansions of Ruin Probability for the Time-Changed fBM Risk Process 23.07.2014 Tianjin, China Hashorva Enkelejd;
Probability Seminar, Nankai University, Mathematical School Individual talk Ruin Probability & Ruin Time Approximation for fBm Risk Model with Tax 21.07.2014 Nankai University, Tianjin, China Hashorva Enkelejd;
International Workshop on Risk Analysis, Ruin and Extremes (see here http://202.113.29.3/2014RARE/) Poster Second-roder asymptotic expansions for products 14.07.2014 Nankai University, Tianjin, China Ling Chengxiu;
International Workshop on Risk Analysis, Ruin and Extremes (see here http://202.113.29.3/2014RARE/) Talk given at a conference Boundary non-crossing probabilities of additive Brownian motion 14.07.2014 Chern Institute, Tianjin, China Hashorva Enkelejd;
International Workshop on Risk Analysis, Ruin and Extremes (see here http://202.113.29.3/2014RARE/) Talk given at a conference Parisian ruin of self-similar Gaussian risk processes 14.07.2014 Chern Institute, Nankai University, Tianjin, China Ji Lanpeng;
The 18th international congress on Insurance: Mathematics and Economics Talk given at a conference Ruin probability of fBm risk processes with tax 10.07.2014 Shanghai, China Ji Lanpeng; Hashorva Enkelejd;
The 18th international congress on Insurance: Mathematics and Economics Talk given at a conference Second order regular variation of products 10.07.2014 Shanghai, China Ling Chengxiu; Hashorva Enkelejd;
The 18th international congress on Insurance: Mathematics and Economics Talk given at a conference Ruin Time Approximation for fBm Risk Models 10.07.2014 Shanghai, China Hashorva Enkelejd; Ji Lanpeng;
Statistical Seminar, Southwest Unviersity, China Individual talk Tail asymptotics expansions of re-insurance risks 02.07.2014 Chongqing, China, China Ling Chengxiu;
Seventh International Workshop on Applied Probability (IWAP 2014) Individual talk Asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals 15.06.2014 Antalya, Turkey Hashorva Enkelejd;
Summer school on the Modelling and Predection of Weather Extremes Poster Tails of Largest Order Statistics 10.06.2014 Annweiler , Germany Ling Chengxiu;
Stochastic Networks And Risk Analysis IV Talk given at a conference Extremes of Chi-processes (E. Hashorva), Extremes of Gamma reflected processes with FBM input (L. Ji) 26.05.2014 Bedlewo, Poland Hashorva Enkelejd; Liu Peng; Engelke Sebastian;
56e Journée de séminaires actuariels Lyon-Lausanne Individual talk Estimation of Brown-Resnick processes based on single extreme events 21.03.2014 Lyon, France Engelke Sebastian;
55e Journée de séminaires actuariels Talk given at a conference Random Scaling and Shifting of Dependent Risks 08.11.2013 Lausanne, Switzerland Hashorva Enkelejd;
Mathematical Seminar, University of Nankai Individual talk Max-stable processes related to Brown-Resnick process 15.07.2013 China, China Engelke Sebastian;
The 8th conference on extreme value analysis, probabilistic and statistical models and their applications Individual talk Second-order tail asymptotics of deflated risks 12.07.2013 Fudan University, China Ling Chengxiu;
8th Conference on Extreme Value Analysis, Probabilistic and Statistical Models Talk given at a conference Estimation of Brown-Resnick processes based on single extreme events 09.07.2013 Fudan University, Shanghai, China Engelke Sebastian;
Mathematisches Kolloquium Talk given at a conference Random Scaling and Shifting: Dependent Risks 04.06.2013 Rostock, Germany, Germany Hashorva Enkelejd;
Mathematisches Kolloquium, Catholic University of Eichstätt-Ingolstadt Talk given at a conference Asymptotic Expansions of Ruin Probability for the Time-Changed fBM Risk Process 17.04.2013 Eichstätt-Ingolstadt, Germany, Germany Hashorva Enkelejd;
Perspectives on Actuarial Risks in Talks of Young Researchers Talk given at a conference Asymptotic Expansions of Ruin Probability for the Time-Changed fBM Risk Process 28.01.2013 Monte Verita, Ascona, Switzerland, Switzerland Hashorva Enkelejd;
Perspectives on Actuarial Risks in Talks of Young Researchers Individual talk Ruin probability for locally self-similar Gaussian processes 27.01.2013 Monte Verita, Ascona, Switzerland, Switzerland Tabis Kamil;
Dept. Mathematical Sciences Colloquium Talk given at a conference Extremes of Gaussian and Related Risks 19.10.2012 Liverpool, Great Britain and Northern Ireland Hashorva Enkelejd;
EAJ Conference Poster On risk processes with two classes of claims and tax 06.09.2012 Lausanne, Switzerland Hashorva Enkelejd;


Awards

Title Year
Prix de Fondation Nicolas et Hélène Porphyrogenis pour la qualité exceptionnelle de son doctorat. 2014

Abstract

The classical Central Limit Theorem and its ramifications show that the Gaussian model is a natural and correct paradigm for building an approximate solution to many otherwise unsolvable problems encountered in various research fields. Indeed, the range of applications of Gaussian processes and related random fields encompasses almost any field of theoretical and applied research. Some extraordinary examplesinclude variations of Brownian motion as the unique solution to problems from theoretical physics, biology, mathematical statistics, risk theory, stochastic finance, telecommunication, just to name a few. While the theory of Gaussian processes and random fields is well-developed and mature, the range of applications of Gaussian random fields is constantly growing. Recently, applications inbrain mapping, cosmology, and quantum chaos have been added to its palmares. Due to the presence of measurement errors, missing observations or random inflations, in some cases the Gaussian framework appears as not tenable. This project advocates that by extending the models to vector-valued chi-processes, vector-valued conditional Gaussian processes and random fields, the Gaussian framework proves to be very reliable.Essentially, numerous applications are intrinsically connected to the study of extremes of Gaussian processes and their related random fields. A natural extreme-value problem in this context is the determination of the exact tail asymptotic behaviour of the maxima of Gaussian processes over some given sets, the hardest and oldest problem in the study of random processes. Besides the tail asymptotics of the maximum, the derivation of limit theorems regarding the maxima of Gaussian processes is both of theoretical and applied interest.This project aims at studying extremes of such large classes of vector-valued Gaussian processes, chi-processes, and conditional Gaussian processes over continuous, discrete and random sets.The principal theoretical findings envisaged by this study shall include both exact tail asymptotic results and limit theorems for the maxima of the mentioned Gaussian and related processes. Since real data are only possible to be observed on a certain discrete grid of time-points,it is planed to investigate the joint asymptotic behaviour of maximum over continuous time intervals with maxima over discrete grids, for several classes of Gaussian processes and chi-processes.Motivated by various applications in risk theory, queueing theory, and hydrodynamics this project is also concerned with the study of the maximum of Gaussian processes and chi-processes over random time intervals. In addition to numerous theoretical results and their interpretation, this project shall develop novel methodologies and techniques. Furthermore, the derivation of some key asymptotical results for the extremes of several Gaussian fields will open the way for novel statistical applications, whereas by focusing on both Gaussian perturbed risk processes and generalsations of the storage processes, additional applications concerned with the risk analysis, simulation of rare-events and the analysis of overflows in hydrodynamics will be promoted.
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