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Swiss National Science Foundation (SNSF)

Wildhainweg 3P.O. Box

CH-3001 Bern

Phone +41 31 308 22 22

English title | Extremes of Gaussian Processes and Related Random Fields |
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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 |

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

Lead |
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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 |

Direct link to Lay Summary | Last update: 21.02.2013 |

Name | Institute |
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Engelke Sebastian | Research Center for Statistics Geneva School of Economics and Management University of Geneva |

Publication |
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A note on ruin problems in perturbed classical risk models |

Aggregation of randomly weighted large risks |

A characterization of the normal distribution using stationary max-stable processes |

A Levy process on the real line seen from its supremum and max-stable processes |

Exact simulation of max-stable processes |

Extremes of -locally stationary Gaussian random fields |

Extremes of a class of non-homogeneous Gaussian random fields |

Extremes of Chi-square Processes with trend |

Extremes of stationary Gaussian storage models |

Extremes on river networks |

Higher-order expansions of distributions of maxima in a Hüsler-Reiss model |

Maxima of skew elliptical triangular arrays |

On maxima of chi-processes over threshold dependent grids |

On Parisian ruin over a finite-time horizon |

Tail asymptotics of generalized deflated risks with insurance applications |

Approximation of a random process with variable smoothness |

Boundary non-crossing probabilities for fractional Brownian motion with trend |

Estimation of Hüsler-Reiss distributions and Brown-Resnick processes |

Extremal Behavior of Gaussian Chaos via Probabilistic Approach |

Extremal behavior of squared Bessel processes attracted by the Brown-Resnick process |

Extremes of Aggregated Dirichlet Risks |

Extremes of homogeneous Gaussian random fields |

Extremes of order statistics of stationary processes |

Extremes of vector-valued Gaussian processes: Exact asymptotics |

Gaussian risk models with financial constraints |

Maxima of a triangular array of multivariate Gaussian sequence |

Max-stable processes and stationary systems of Levy particles |

On Laplace asymptotic method, with application to random chaos |

On Sarmanov Mixed Erlang Risks in Insurance Applications |

On the gamma-reflected processes with fBm input |

Parisian ruin of self-similar Gaussian risk processes |

Piterbarg theorems for chi-processes with trend |

Piterbarg's max-discretisation theorem for stationary vector Gaussian processes observed on different grids |

Tail approximation for reinsurance portfolios of Gaussian-like risks |

Tail Behaviour of Weighted Sums of Order Statistics of Dependent Risks |

Tail dependence for two skew slash distributions |

Tail dependence for two skew slash distributions |

Tail asymptotic expansions for L-statisitcs. |

A duality result for the generalized Erlang risk model |

Aggregation of log-linear risks |

Approximation of passage times of gamma-reflected processes with fBm input |

Asymptotics for a Discrete-time Risk Model with the Emphasis on Financial Risk |

Asymptotics of the finite-time ruin probability for the Sparre Andersen risk model perturbed by an inflated stationary chi-process |

Berman's inequality under random scaling |

Boundary Non-Crossings of Additive Wiener Fields |

EXTREMES AND FIRST PASSAGE TIMES OF CORRELATED FRACTIONAL BROWNIAN MOTIONS |

Extremes of perturbed bivariate Rayleigh risks |

Finite-time ruin probability of aggregate Gaussian processes |

Gaussian approximation of perturbed chi-square risks |

Limit properties of exceedances point processes of scaled stationary Gaussian sequences |

Maxima and minima of complete and incomplete stationary sequences |

Modeling of Censored Bivariate Extremal Events |

On Piterbarg max-discretisation theorem for standardised maximum of stationary gaussian processes |

On the probability of conjunctions of stationary Gaussian processes |

Random shifting and scaling of insurance risks |

Second order asymptotics of aggregated log-elliptical risk |

Tail asymptotics of random sum and maximum of log-normal risks |

Tail asymptotics of supremum of certain Gaussian processes over threshold dependent random intervals |

Large deviations of Shepp statistics for fractional Brownian motion |

Aggregation of parametrised log-elliptical risks |

ECOMOR and LCR reinsurance with gamma-like claims |

Efficient simulation of tail probabilities for sums of log-elliptical risks |

Exact asymptotics and limit theorems for supremum of stationary chi-processes over a random interval. |

Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval |

Limit theorems for extremes of strongly dependent cyclo-stationary chi-processes |

On Extremal Behavior of Gaussian Chaos |

On Piterbarg theorem for the maxima of stationary Gaussian sequences |

On Piterbarg's max-discretisation theorem for multivariate stationary Gaussian processes |

On the supremum of gamma-reflected processes with fractional Brownian motion as input |

Asymptotics of maxima of strongly dependent Gaussian processes |

Calculation of Bayes premium for conditionally elliptical risks |

Group / person | Country |
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Types of collaboration |
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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 |

Title | Type of contribution | Title of article or contribution | Date | Place | Persons involved |
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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; |

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

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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CH-3001 Bern

Phone +41 31 308 22 22

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