Operational Risk and Pareto Lévy Copulas

Size: px
Start display at page:

Download "Operational Risk and Pareto Lévy Copulas"

Transcription

1 Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR - a closed form approximation. Risk, December Böcker, K. and Klüppelberg, C. (28) First order approximations to operational risk - dependence and consequences. To appear in: G.N. Gregoriou (ed.) Operational Risk Toward Basel III, Best Practices and Issues in Modeling, Management and Regulation. Wiley, New York.

2 Contents (1) Basel II (2) The Loss Distribution Approach (LDA) for the Single Cell (3) Modelling Dependence of Lévy Processes (4) The Multivariate Subexponential Compound Poisson (SCP) Model (5) Estimating Total OpVar

3 (1) Basel II Structure of Risk Management: Pillar 1: minimal capital requirements Pillar 2: supervisory review of capital adequacy Pillar 3: public disclosure Definition of Operational Risk: The risk of losses resulting from inadequate or failed internal processes, people and systems, or external events.

4 Examples: 1995 Barings Bank: Nick Leeson (1.3b British Pounds) 21 Enron (largest US bankruptcy ever) 25 Brokerhouse Mizuho Securities: instead of selling 1 share for 61 Yen a trader wrote 61 shares for 1 Yen each (19 Mio Euro) 28 Societé Générale: Jerome Kerviel (4.9b Euro) Basel II distinguishes 7 loss types and 8 business types

5 (2) The Loss Distribution Approach (LDA) for the Single Cell Subexponential compound Poisson (SCP) model (1) The severities (X k ) k N are positive iid random variables with subexponential distribution function F. (2) The frequency process N(t) of loss events in the time interval [,t] for t constitutes a homogenous Poisson process with intensity λ >. In particular, λt (λt)n P(N(t) = n) = p t (n) = e, n N. n! (3) The severity process and the frequency process are independent. (4) The aggregate loss process is given by S(t) = N(t) k=1 X k, t.

6 Let (X k ) k N be iid random variables with distribution function F. Then F is said to be subexponential (F S) if lim x P(X X n > x) P(max(X 1,...,X n ) > x) = 1 for some (all) n 2. Denote F(x) := 1 F(x) = P(X > x) for x > the distribution tail of F. If for some α, lim t F(xt) F(t) = x α, x >, then F is called regularly varying with index α, denoted by F R α. The quantity α is also called the tail index of F or X. See Embrechts, Klüppelberg and Mikosch (1997) for details.

7 Theorem [Analytical OpVaR] Consider the SCP model for fixed t > and a subexponential severity with distribution function F. (a) Assume that F R α for α (, ). Then as κ 1, ( ) ( ) ( ) ( ) 1 λt 1 λ VaR t (κ) t 1/α F 1 κ F 1 κ ( ) 1/α ( ) λt λt = t 1/α VaR 1 (κ) L. 1 κ 1 κ (b) If 1/F R α for α (, ] (α = means that F(t)/F(xt) for x > 1 and F(t)/F(xt) for x < 1), then VaR t (κ) ( 1 F ) ( ) λt, κ 1. 1 κ

8 Popular subexponential severity distributions Name Distribution function Parameters Lognormal ( ) lnx µ F(x) = Φ σ µ R, σ > Weibull F(x) = 1 e (x/θ)τ θ >, < τ < 1 Pareto ( F(x) = x α θ) α,θ >

9 Approximated VaR (dashed line) and simulated VaR (solid line) for the Pareto-Poisson LDA with θ = 1. 5 alpha = alpha = 1.1 VaR 4 3 VaR Confidence Level Confidence Level

10 (3) Modelling Dependence of Lévy Processes Invoking the copula idea: A d-dimensional copula C is a distribution function on [,1] d with standard uniform marginals. Theorem [Sklar s Theorem] Let F be a joint distribution function with marginals F 1,...,F d. Then there exists a copula C : [, 1] d [, 1] such that for all x 1,...,x d R = [, ] F(x 1,...,x d ) = C(F 1 (x 1 ),...,F d (x d )). (1) If the marginals are continuous, then C is unique. Otherwise it is unique on Ran F 1 Ran F d. Conversely, if C is a copula and F 1,...,F d are distribution functions, then the function F as defined in (1) is a joint distribution function with marginals F 1,...,F d.

11 Question Can we use copulas to model dependence of Lévy processes? Our LDA model is a compound Poisson process with positive jumps. Problems The law of a Lévy process X is completely determined by the distribution of X at time t for any t >. The copula C t of (X 1 (t),...,x d (t)) may depend on t. In general, C s cannot be calculated from C t, because C s depends also on the marginal distributions. For given infinitely divisible marginal distributions it is unclear, which copulas C t yield multivariate infinite divisible distributions. (Copulas are invariant under strictly increasing transformations, infinite divisibility is not!) Introduce a Lévy copula [Cont & Tankov (24), Kallsen & Tankov (24), Barndorff-Nielsen & Lindner (24).

12 Problem Lévy measures may have a non-integrable singularity at. Define E := [, ] d \ {}. Let X be a spectrally positive Lévy process in R d with a Lévy measure Γ, which has standard 1-stable one-dimensional marginals (Γ i (x) = x 1 for x > ). Then we call Γ a Pareto Lévy measure and the associated tail measure Γ(x) = Γ([x 1, ) [x d, )) =: Ĉ(x 1,...,x d ), x E, is called Pareto Lévy copula Ĉ. Remark Extension to general Lévy processes by quadrantwise definition (singularity in!).

13 Lemma Let X be a spectrally positive Lévy process in R d with Lévy measure Π on E and continuous marginal tail measures Π 1,...,Π d, where Π i (x) := Π([x, )) for i = 1,...,d. Then ( Π(x) = Π([x 1, ] [x d, ]) = Ĉ 1 Π 1 (x 1 ),..., 1 ) Π d (x d ), x E, and Ĉ is a Pareto Lévy copula.

14 Theorem [Sklar s Theorem for Pareto Lévy copulas] Let Π be the tail measure of a d-dimensional spectrally positive Lévy process with marginal tail measures Π 1,...,Π d. Then there exists a Pareto Lévy copula Ĉ : E [, ] such that ( Π(x) = Ĉ 1 Π 1 (x 1 ),..., 1 Π d (x d ) If the marginal tail measures are ( continuous on [, ( ], then 1 1 Otherwise, it is unique on Ran Ran. Π 1 ) Π d ) ). (2) Ĉ is unique. Conversely, if Ĉ is a Pareto Lévy copula and Π 1,...,Π d are marginal tail measures, then Π as defined in (2) is a joint tail measure with marginals Π 1,...,Π d.

15 Examples of Pareto Lévy copulas Example Clayton Pareto Lévy copula (special Archimedian Pareto Lévy copula) Ĉ ϑ (x 1,...,x d ) = (x 1 ϑ + + x d ϑ ) 1/ϑ Note: lim ϑ Ĉ θ (x 1,...,x d ) = Ĉ (x 1,...,x d ) complete positive dependence, lim ϑ Ĉ ϑ (x 1,...,x d ) = Ĉ (x 1,...,x d ) independence. A Pareto Lévy copula Ĉ is homogeneous (of order 1), if for all t > Ĉ(x 1,...,x d ) = t Ĉ(tx 1,...,tx d ), (x 1,...,x d ) E.

16 θ =.3 1/2 stable severities, Clayton Levy copula: θ=.3 Jumps: 12.6 X X, Y t Y t t θ =2 1/2 stable severities, Clayton Levy copula: Jumps: θ= X X, Y t.2.5 Y t t θ =1 1/2 stable severities, Clayton Levy copula: θ=1 Jumps:.5.15 X t Y X, Y t t

17 (4) The Multivariate SCP Model (1) All operational risk cells, indexed by i = 1,...,d, are described by an SCP model with aggregate loss process S i, continuous subexponential severity distribution function F i and Poisson parameter λ i >. (2) Dependence between different cells is modelled by a Pareto Lévy copula: Let Π i : [, ) [, ) be the tail measure to S i, i.e. Π i ( ) = λ i F i ( ), and let Ĉ : E [, ] be a Pareto Lévy copula. Then ( ) Π(x 1,...,x d ) = Ĉ 1,..., 1 λ 1 F 1 (x 1 ) λ d F d (x d ) defines the tail measure of the compound Poisson process S = (S 1,...,S d ). (3) The bank s total aggregate operational loss process is defined as with tail measure S + (t) = S 1 (t) + S 2 (t) + + S d (t), t, Π + (z) = Π({(x 1,...,x d ) [, ) d : d i=1 x i z}), z.

18 Proposition Consider the multivariate SCP model. Its total aggregate loss process S + is compound Poisson with intensity λ + = lim z Π + (z) and severity distribution F + (z) = 1 F + (z) = Π+ (z) λ +, z. Consider the multivariate SCP model with total aggregate loss S + (t) at time t > and denote G + t ( ) = P(S + (t) ). Total Operational VaR up to time t at confidence level κ is defined as VaR + t (κ) = G + t (κ), κ (,1), with G + t (κ) = inf{z R : G + t (z) κ} for < κ < 1.

19 (5) Estimating Total OpVar One cell dominant Theorem For fixed t > let S i (t) for i = 1,...,d have compound Poisson distributions. Assume that F 1 R α for α >. Let ρ > α and suppose that E[(X i ) ρ ] < for i = 2,...,d. Then regardless of the dependence structure between (S 1 (t),...,s d (t)), P(S 1 (t) + + S d (t) > x) EN 1 (t) P(X 1 > x), x, ( VaR + t (κ) F1 1 1 κ ) = VaR 1 EN 1 (t) t(κ), κ 1.

20 Multivariate SCP model with completely dependent cells - All cell processes jump together = λ := λ 1 = = λ d. - The mass of the Lévy measure is concentrated on {(x 1,...,x d ) (, ) d : Π 1 (x 1 ) = = Π d (x d )} = {(x 1,...,x d ) (, ) d : F 1 (x 1 ) = = F d (x d )}. Let F i be strictly increasing and continuous: F 1 i (q) exists for all q [, 1). Then Π + (z) = Π({(x 1,...,x d ) (, ) d : = Π 1 ({x 1 (, ) : x 1 + d i=2 d x i z}) i=1 F 1 i (F 1 (x 1 )) z}), z >.

21 Set H(x 1 ) := x 1 + d invertible. Thus i=2 F 1 i (F 1 (x 1 )) for x 1 (, ) and note that it is Π + ( (z) = Π 1 ({x 1 (, ) : x 1 H 1 (z)}) = Π 1 H 1 (z) ), z >. Theorem Consider a multivariate SCP model with completely dependent cell processes S 1,...,S d and strictly increasing and continuous severity distributions F i. Then, S + is compound Poisson with parameters λ + = λ and F + ( (z) = F 1 H 1 (z) ). If F + S R α for α (, ], then VaR + t (κ) d VaR i t(κ), κ 1, i=1 where VaR i t( ) denotes the stand alone OpVaR of cell i.

22 Corollary Assume that the conditions of the Theorem hold and that F 1 R α for α (, ) and F i (x) lim x F 1 (x) = c i [, ). Assume that c i for i = 1,...,b d and c i = for i = b + 1,...,d. Then VaR + t (κ) b i=1 c 1/α i VaR 1 t(κ), κ 1.

23 Multivariate SCP model with independent cells - Not two cell processes ever jump together. - The mass of the Lévy measure is concentrated on the axes. Π + (z) = Π 1 (z 1 ) + + Π d (z d ). Theorem Assume S 1,...,S d are independent. Then S + defines a one-dimensional SCP model with parameters λ + = λ λ d and F + (z) = 1 λ + [ λ1 F 1 (z) + + λ d F d (z) ], z. If F 1 R α for α (, ) and for all i = 2,...,d, lim x F i (x) F 1 (x) = c i [, ), then, setting C λ = λ 1 + c 2 λ c d λ d, VaR + t (κ) ( 1 F 1 ) ( ) Cλ t 1 κ = VaR 1 t ( ) Cλ t 1 κ, κ 1.

24 Multivariate SCP model with regularly varying Lévy measure (a) Let Π be a Lévy measure of a spectrally positive Lévy process in R d. Assume that there exists a Radon measure ν on E such that for x E lim u Π({y : y 1 > ux 1 or or y d > ux d } Π 1 (u) = ν({y : y 1 > x 1 or or y d > x d }) =: ν([,x] c ). Then we call Π multivariate regularly varying. (b) The measure ν has a scaling property: there exists some α > such that for every s > ν([, sx] c ) = s α ν([,x] c ), and Π is called multivariate regularly varying with index α.

25 Some result for multivariate regularly varying and stable processes Theorem Let the spectrally positive Lévy process X in R d have tail measure Π. Then Π is multivariate regularly varying with index α > if and only if (1) at least one marginal tail measure Π i R α for α > and all other marginal tail measures have the same order or are lighter, and (2) Γ is multivariate regularly varying with index 1. Theorem [Kallsen and Tankov (26)] For < α < 2 let X be a Lévy process in R d. The process X is α-stable if and only if it has α-stable one-dimensional marginals and it has a Pareto Lévy copula, which is homogeneous of order 1.

26 Theorem Consider an SCP model with multivariate regularly varying cell processes (S 1,...,S d ) with index α and limit measure ν. Assume further that the severity distributions F i for i = 1,...,d are strictly increasing and continuous. Then, S + is compound Poisson with parameters λ + F + (x) ν + (1, )λ 1 F 1 (x), x. where ν + (z, ] = ν{x : d i=1 x i > z} for z >. Furthermore, λ + F + ( ) R α and total OpVaR is asymptotically given by VaR t (κ) F 1 ( 1 ) 1 κ t λ 1 ν + (1, ], κ 1.

27 Example [Clayton Pareto Lévy copula] Assume that F 2 (x)/f 1 (x) c as x. Set c := (λ 2 /λ 1 )c. [( ν + (1, ] = 1 + c 1/α E c 1/α + Y 1/α ϑ ) α 1)], where Y ϑ is a positive random variable with density g(s) = (1 + s ϑ ) 1/ϑ 1. For α = 1 we have ν + (1, ] = 1 + c, independent of ϑ. Consequently, total OpVar is for all < ϑ < asymptotically equal to the independent OpVar. If αϑ = 1, then ν + (1, ] = c1+1/α 1 c 1/α 1

28 References Böcker, K. and Klüppelberg, C. (28) First order approximation to operational risk - dependence and consequences. In: G.N. Gregoriou (ed.), Operational Risk Towards Basel III, Best Practices and Issues in Modeling, Management and Regulation. Wiley, New York. To appear. Böcker, K. and Klüppelberg, C. (28) Modelling and measuring multivariate operational risk with Lévy copulas J. Operational Risk 3(2), Böcker, K. and Klüppelberg, C. (28) Economic Capital Modelling and Basel II Compliance in the Banking Industry. In: Jger, W. and Krebs, H.-J. (Eds.) Mathematics - Key Technology for the Future. Springer, Berlin.

29 Bregman, Y. and Klüppelberg, C. (25) Ruin estimation in multivariate models with Clayton dependence structure. Scand. Act. J. 25(6), Eder, I. and Klüppelberg, C. (27) The quintuple law for sums of dependent Lévy processes. Submitted for publication. Eder, I. and Klüppelberg, C. (28) Pareto Lévy copulas and multivariate regular variation. In preparation. Technische Universität München. Esmaeili, H. and Klüppelberg, C. (28) Parameter estimation of a bivariate compound Poisson process. Submitted for publication. Klüppelberg, C. and Resnick, S. (28) The Pareto copula, aggregation of risks and the Emperor s socks. J. Appl. Prob. 45 (1),

Operational Risk and Pareto Lévy Copulas

Operational Risk and Pareto Lévy Copulas Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR

More information

Multivariate Models for Operational Risk

Multivariate Models for Operational Risk Multivariate Models for Operational Risk Klaus Böcker Claudia Klüppelberg Abstract In Böcker and Klüppelberg 25) we presented a simple approximation of Op- VaR of a single operational risk cell. The present

More information

Multivariate Operational Risk: Dependence Modelling with Lévy Copulas

Multivariate Operational Risk: Dependence Modelling with Lévy Copulas Multivariate Operational Risk: Dependence Modelling with Lévy Copulas Klaus Böcker Claudia Klüppelberg Abstract Simultaneous modelling of operational risks occurring in different event type/business line

More information

Parametric estimation of a bivariate stable Lévy process

Parametric estimation of a bivariate stable Lévy process Parametric estimation of a bivariate stable Lévy process Habib Esmaeili Claudia Klüppelberg January 3, 20 Abstract We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula

More information

Parameter estimation of a bivariate compound Poisson process

Parameter estimation of a bivariate compound Poisson process Parameter estimation of a bivariate compound Poisson process Habib Esmaeili Claudia Klüppelberg August 5, Abstract In this article, we review the concept of a Lévy copula to describe the dependence structure

More information

Parameter estimation of a Lévy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling

Parameter estimation of a Lévy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling Parameter estimation of a Lévy copula of a discretely observed bivariate compound Poisson process with an application to operational risk modelling J. L. van Velsen 1,2 arxiv:1212.0092v1 [q-fin.rm] 1 Dec

More information

Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications

Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010 1. Preliminaries

More information

Quantitative Modeling of Operational Risk: Between g-and-h and EVT

Quantitative Modeling of Operational Risk: Between g-and-h and EVT : Between g-and-h and EVT Paul Embrechts Matthias Degen Dominik Lambrigger ETH Zurich (www.math.ethz.ch/ embrechts) Outline Basel II LDA g-and-h Aggregation Conclusion and References What is Basel II?

More information

Overview of Extreme Value Theory. Dr. Sawsan Hilal space

Overview of Extreme Value Theory. Dr. Sawsan Hilal space Overview of Extreme Value Theory Dr. Sawsan Hilal space Maths Department - University of Bahrain space November 2010 Outline Part-1: Univariate Extremes Motivation Threshold Exceedances Part-2: Bivariate

More information

Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims

Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Proceedings of the World Congress on Engineering 29 Vol II WCE 29, July 1-3, 29, London, U.K. Finite-time Ruin Probability of Renewal Model with Risky Investment and Subexponential Claims Tao Jiang Abstract

More information

Ruin probabilities in multivariate risk models with periodic common shock

Ruin probabilities in multivariate risk models with periodic common shock Ruin probabilities in multivariate risk models with periodic common shock January 31, 2014 3 rd Workshop on Insurance Mathematics Universite Laval, Quebec, January 31, 2014 Ruin probabilities in multivariate

More information

On the Conditional Value at Risk (CoVaR) from the copula perspective

On the Conditional Value at Risk (CoVaR) from the copula perspective On the Conditional Value at Risk (CoVaR) from the copula perspective Piotr Jaworski Institute of Mathematics, Warsaw University, Poland email: P.Jaworski@mimuw.edu.pl 1 Overview 1. Basics about VaR, CoVaR

More information

VaR vs. Expected Shortfall

VaR vs. Expected Shortfall VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004) Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures

More information

Copulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E

Copulas. Mathematisches Seminar (Prof. Dr. D. Filipovic) Di Uhr in E Copulas Mathematisches Seminar (Prof. Dr. D. Filipovic) Di. 14-16 Uhr in E41 A Short Introduction 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 The above picture shows a scatterplot (500 points) from a pair

More information

Operational risk modeled analytically II: the consequences of classification invariance. Vivien BRUNEL

Operational risk modeled analytically II: the consequences of classification invariance. Vivien BRUNEL Operational risk modeled analytically II: the consequences of classification invariance Vivien BRUNEL Head of Risk and Capital Modeling, Société Générale and Professor of Finance, Léonard de Vinci Pôle

More information

Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness

Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness Additivity properties for Value-at-Risk under Archimedean dependence and heavy-tailedness Paul Embrechts, Johanna Nešlehová, Mario V. Wüthrich Abstract Mainly due to new capital adequacy standards for

More information

Lévy copulas: review of recent results

Lévy copulas: review of recent results Lévy copulas: review of recent results Peter Tankov Abstract We review and extend the now considerable literature on Lévy copulas. First, we focus on Monte Carlo methods and present a new robust algorithm

More information

Modelling Dependent Credit Risks

Modelling Dependent Credit Risks Modelling Dependent Credit Risks Filip Lindskog, RiskLab, ETH Zürich 30 November 2000 Home page:http://www.math.ethz.ch/ lindskog E-mail:lindskog@math.ethz.ch RiskLab:http://www.risklab.ch Modelling Dependent

More information

Simulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris

Simulating Exchangeable Multivariate Archimedean Copulas and its Applications. Authors: Florence Wu Emiliano A. Valdez Michael Sherris Simulating Exchangeable Multivariate Archimedean Copulas and its Applications Authors: Florence Wu Emiliano A. Valdez Michael Sherris Literatures Frees and Valdez (1999) Understanding Relationships Using

More information

Modelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich

Modelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich Modelling Dependence with Copulas and Applications to Risk Management Filip Lindskog, RiskLab, ETH Zürich 02-07-2000 Home page: http://www.math.ethz.ch/ lindskog E-mail: lindskog@math.ethz.ch RiskLab:

More information

Risk Aggregation and Model Uncertainty

Risk Aggregation and Model Uncertainty Risk Aggregation and Model Uncertainty Paul Embrechts RiskLab, Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/ embrechts/ Joint work with A. Beleraj, G. Puccetti and L. Rüschendorf

More information

arxiv: v1 [math.st] 7 Mar 2015

arxiv: v1 [math.st] 7 Mar 2015 March 19, 2018 Series representations for bivariate time-changed Lévy models arxiv:1503.02214v1 [math.st] 7 Mar 2015 Contents Vladimir Panov and Igor Sirotkin Laboratory of Stochastic Analysis and its

More information

Randomly Weighted Sums of Conditionnally Dependent Random Variables

Randomly Weighted Sums of Conditionnally Dependent Random Variables Gen. Math. Notes, Vol. 25, No. 1, November 2014, pp.43-49 ISSN 2219-7184; Copyright c ICSRS Publication, 2014 www.i-csrs.org Available free online at http://www.geman.in Randomly Weighted Sums of Conditionnally

More information

of Lévy copulas. dynamics and transforms of Upsilon-type

of Lévy copulas. dynamics and transforms of Upsilon-type Lévy copulas: dynamics and transforms of Upsilon-type Ole E. Barndorff-Nielsen Alexander M. Lindner Abstract Lévy processes and infinitely divisible distributions are increasingly defined in terms of their

More information

THIELE CENTRE for applied mathematics in natural science

THIELE CENTRE for applied mathematics in natural science THIELE CENTRE for applied mathematics in natural science Tail Asymptotics for the Sum of two Heavy-tailed Dependent Risks Hansjörg Albrecher and Søren Asmussen Research Report No. 9 August 25 Tail Asymptotics

More information

SOME INDICES FOR HEAVY-TAILED DISTRIBUTIONS

SOME INDICES FOR HEAVY-TAILED DISTRIBUTIONS SOME INDICES FOR HEAVY-TAILED DISTRIBUTIONS ROBERTO DARIS and LUCIO TORELLI Dipartimento di Matematica Applicata B. de Finetti Dipartimento di Scienze Matematiche Universitk degli Studi di Trieste P.le

More information

Approximating the Integrated Tail Distribution

Approximating the Integrated Tail Distribution Approximating the Integrated Tail Distribution Ants Kaasik & Kalev Pärna University of Tartu VIII Tartu Conference on Multivariate Statistics 26th of June, 2007 Motivating example Let W be a steady-state

More information

Risk Aggregation. Paul Embrechts. Department of Mathematics, ETH Zurich Senior SFI Professor.

Risk Aggregation. Paul Embrechts. Department of Mathematics, ETH Zurich Senior SFI Professor. Risk Aggregation Paul Embrechts Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/~embrechts/ Joint work with P. Arbenz and G. Puccetti 1 / 33 The background Query by practitioner

More information

Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation

Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation Randomly weighted sums under a wide type of dependence structure with application to conditional tail expectation Shijie Wang a, Yiyu Hu a, Lianqiang Yang a, Wensheng Wang b a School of Mathematical Sciences,

More information

Heavy Tailed Time Series with Extremal Independence

Heavy Tailed Time Series with Extremal Independence Heavy Tailed Time Series with Extremal Independence Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Herold Dehling Bochum January 16, 2015 Rafa l Kulik and Philippe Soulier Regular variation

More information

CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS

CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS EVA IV, CONTAGION VERSUS FLIGHT TO QUALITY IN FINANCIAL MARKETS Jose Olmo Department of Economics City University, London (joint work with Jesús Gonzalo, Universidad Carlos III de Madrid) 4th Conference

More information

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables

Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables Asymptotic Tail Probabilities of Sums of Dependent Subexponential Random Variables Jaap Geluk 1 and Qihe Tang 2 1 Department of Mathematics The Petroleum Institute P.O. Box 2533, Abu Dhabi, United Arab

More information

Ruin, Operational Risk and How Fast Stochastic Processes Mix

Ruin, Operational Risk and How Fast Stochastic Processes Mix Ruin, Operational Risk and How Fast Stochastic Processes Mix Paul Embrechts ETH Zürich Based on joint work with: - Roger Kaufmann (ETH Zürich) - Gennady Samorodnitsky (Cornell University) Basel Committee

More information

Tail Approximation of Value-at-Risk under Multivariate Regular Variation

Tail Approximation of Value-at-Risk under Multivariate Regular Variation Tail Approximation of Value-at-Risk under Multivariate Regular Variation Yannan Sun Haijun Li July 00 Abstract This paper presents a general tail approximation method for evaluating the Valueat-Risk of

More information

Regular Variation and Extreme Events for Stochastic Processes

Regular Variation and Extreme Events for Stochastic Processes 1 Regular Variation and Extreme Events for Stochastic Processes FILIP LINDSKOG Royal Institute of Technology, Stockholm 2005 based on joint work with Henrik Hult www.math.kth.se/ lindskog 2 Extremes for

More information

Lecture Quantitative Finance Spring Term 2015

Lecture Quantitative Finance Spring Term 2015 on bivariate Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 07: April 2, 2015 1 / 54 Outline on bivariate 1 2 bivariate 3 Distribution 4 5 6 7 8 Comments and conclusions

More information

Risk Aggregation with Dependence Uncertainty

Risk Aggregation with Dependence Uncertainty Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich

More information

Modèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine

Modèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine Séminaire de Statistiques de l'irma Modèles de dépendance entre temps inter-sinistres et montants de sinistre en théorie de la ruine Romain Biard LMB, Université de Franche-Comté en collaboration avec

More information

Characterization of dependence of multidimensional Lévy processes using Lévy copulas

Characterization of dependence of multidimensional Lévy processes using Lévy copulas Characterization of dependence of multidimensional Lévy processes using Lévy copulas Jan Kallsen Peter Tankov Abstract This paper suggests to use Lévy copulas to characterize the dependence among components

More information

Asymptotics of sums of lognormal random variables with Gaussian copula

Asymptotics of sums of lognormal random variables with Gaussian copula Asymptotics of sums of lognormal random variables with Gaussian copula Søren Asmussen, Leonardo Rojas-Nandayapa To cite this version: Søren Asmussen, Leonardo Rojas-Nandayapa. Asymptotics of sums of lognormal

More information

Dependence. MFM Practitioner Module: Risk & Asset Allocation. John Dodson. September 11, Dependence. John Dodson. Outline.

Dependence. MFM Practitioner Module: Risk & Asset Allocation. John Dodson. September 11, Dependence. John Dodson. Outline. MFM Practitioner Module: Risk & Asset Allocation September 11, 2013 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y

More information

Markov Switching Regular Vine Copulas

Markov Switching Regular Vine Copulas Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS057) p.5304 Markov Switching Regular Vine Copulas Stöber, Jakob and Czado, Claudia Lehrstuhl für Mathematische Statistik,

More information

The extremal elliptical model: Theoretical properties and statistical inference

The extremal elliptical model: Theoretical properties and statistical inference 1/25 The extremal elliptical model: Theoretical properties and statistical inference Thomas OPITZ Supervisors: Jean-Noel Bacro, Pierre Ribereau Institute of Mathematics and Modeling in Montpellier (I3M)

More information

Chapter 2 Asymptotics

Chapter 2 Asymptotics Chapter Asymptotics.1 Asymptotic Behavior of Student s Pdf Proposition.1 For each x R d,asν, f ν,,a x g a, x..1 Proof Let a = 0. Using the well-known formula that Ɣz = π z e z z z 1 + O 1, as z,. z we

More information

Tail Dependence of Multivariate Pareto Distributions

Tail Dependence of Multivariate Pareto Distributions !#"%$ & ' ") * +!-,#. /10 243537698:6 ;=@?A BCDBFEHGIBJEHKLB MONQP RS?UTV=XW>YZ=eda gihjlknmcoqprj stmfovuxw yy z {} ~ ƒ }ˆŠ ~Œ~Ž f ˆ ` š œžÿ~ ~Ÿ œ } ƒ œ ˆŠ~ œ

More information

Extreme Value Analysis and Spatial Extremes

Extreme Value Analysis and Spatial Extremes Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models

More information

Behaviour of multivariate tail dependence coefficients

Behaviour of multivariate tail dependence coefficients ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 22, Number 2, December 2018 Available online at http://acutm.math.ut.ee Behaviour of multivariate tail dependence coefficients Gaida

More information

Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution

Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Daniel Alai Zinoviy Landsman Centre of Excellence in Population Ageing Research (CEPAR) School of Mathematics, Statistics

More information

Exponential functionals of Lévy processes

Exponential functionals of Lévy processes Exponential functionals of Lévy processes Víctor Rivero Centro de Investigación en Matemáticas, México. 1/ 28 Outline of the talk Introduction Exponential functionals of spectrally positive Lévy processes

More information

GARCH processes continuous counterparts (Part 2)

GARCH processes continuous counterparts (Part 2) GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/

More information

Importance Sampling Methodology for Multidimensional Heavy-tailed Random Walks

Importance Sampling Methodology for Multidimensional Heavy-tailed Random Walks Importance Sampling Methodology for Multidimensional Heavy-tailed Random Walks Jose Blanchet (joint work with Jingchen Liu) Columbia IEOR Department RESIM Blanchet (Columbia) IS for Heavy-tailed Walks

More information

Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties

Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties Characterization through Hazard Rate of heavy tailed distributions and some Convolution Closure Properties Department of Statistics and Actuarial - Financial Mathematics, University of the Aegean October

More information

Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables

Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables Walter Schneider July 26, 20 Abstract In this paper an analytic expression is given for the bounds

More information

Multivariate Markov-switching ARMA processes with regularly varying noise

Multivariate Markov-switching ARMA processes with regularly varying noise Multivariate Markov-switching ARMA processes with regularly varying noise Robert Stelzer 26th February 2007 Abstract The tail behaviour of stationary R d -valued Markov-Switching ARMA processes driven

More information

Rare Events in Random Walks and Queueing Networks in the Presence of Heavy-Tailed Distributions

Rare Events in Random Walks and Queueing Networks in the Presence of Heavy-Tailed Distributions Rare Events in Random Walks and Queueing Networks in the Presence of Heavy-Tailed Distributions Sergey Foss Heriot-Watt University, Edinburgh and Institute of Mathematics, Novosibirsk The University of

More information

Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims

Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims Reinsurance and ruin problem: asymptotics in the case of heavy-tailed claims Serguei Foss Heriot-Watt University, Edinburgh Karlovasi, Samos 3 June, 2010 (Joint work with Tomasz Rolski and Stan Zachary

More information

WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES AND THEIR MAXIMA

WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES AND THEIR MAXIMA Adv. Appl. Prob. 37, 510 522 2005 Printed in Northern Ireland Applied Probability Trust 2005 WEIGHTED SUMS OF SUBEXPONENTIAL RANDOM VARIABLES AND THEIR MAXIMA YIQING CHEN, Guangdong University of Technology

More information

Practical conditions on Markov chains for weak convergence of tail empirical processes

Practical conditions on Markov chains for weak convergence of tail empirical processes Practical conditions on Markov chains for weak convergence of tail empirical processes Olivier Wintenberger University of Copenhagen and Paris VI Joint work with Rafa l Kulik and Philippe Soulier Toronto,

More information

Tail dependence coefficient of generalized hyperbolic distribution

Tail dependence coefficient of generalized hyperbolic distribution Tail dependence coefficient of generalized hyperbolic distribution Mohalilou Aleiyouka Laboratoire de mathématiques appliquées du Havre Université du Havre Normandie Le Havre France mouhaliloune@gmail.com

More information

Estimation of the long Memory parameter using an Infinite Source Poisson model applied to transmission rate measurements

Estimation of the long Memory parameter using an Infinite Source Poisson model applied to transmission rate measurements of the long Memory parameter using an Infinite Source Poisson model applied to transmission rate measurements François Roueff Ecole Nat. Sup. des Télécommunications 46 rue Barrault, 75634 Paris cedex 13,

More information

THIELE CENTRE. Markov Dependence in Renewal Equations and Random Sums with Heavy Tails. Søren Asmussen and Julie Thøgersen

THIELE CENTRE. Markov Dependence in Renewal Equations and Random Sums with Heavy Tails. Søren Asmussen and Julie Thøgersen THIELE CENTRE for applied mathematics in natural science Markov Dependence in Renewal Equations and Random Sums with Heavy Tails Søren Asmussen and Julie Thøgersen Research Report No. 2 June 216 Markov

More information

Multivariate Distribution Models

Multivariate Distribution Models Multivariate Distribution Models Model Description While the probability distribution for an individual random variable is called marginal, the probability distribution for multiple random variables is

More information

Asymptotics for Risk Capital Allocations based on Conditional Tail Expectation

Asymptotics for Risk Capital Allocations based on Conditional Tail Expectation Asymptotics for Risk Capital Allocations based on Conditional Tail Expectation Alexandru V. Asimit Cass Business School, City University, London EC1Y 8TZ, United Kingdom. E-mail: asimit@city.ac.uk Edward

More information

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.

Dependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline. Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,

More information

On the Estimation and Application of Max-Stable Processes

On the Estimation and Application of Max-Stable Processes On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang

More information

Relations Between Hidden Regular Variation and Tail Order of. Copulas

Relations Between Hidden Regular Variation and Tail Order of. Copulas Relations Between Hidden Regular Variation and Tail Order of Copulas Lei Hua Harry Joe Haijun Li December 28, 2012 Abstract We study the relations between tail order of copulas and hidden regular variation

More information

Poisson Cluster process as a model for teletraffic arrivals and its extremes

Poisson Cluster process as a model for teletraffic arrivals and its extremes Poisson Cluster process as a model for teletraffic arrivals and its extremes Barbara González-Arévalo, University of Louisiana Thomas Mikosch, University of Copenhagen Gennady Samorodnitsky, Cornell University

More information

arxiv: v1 [math.pr] 10 Dec 2009

arxiv: v1 [math.pr] 10 Dec 2009 The Annals of Applied Probability 29, Vol. 19, No. 6, 247 279 DOI: 1.1214/9-AAP61 c Institute of Mathematical Statistics, 29 arxiv:912.1925v1 [math.pr] 1 Dec 29 THE FIRST PASSAGE EVENT FOR SUMS OF DEPENDENT

More information

Calculating credit risk capital charges with the one-factor model

Calculating credit risk capital charges with the one-factor model Calculating credit risk capital charges with the one-factor model Susanne Emmer Dirk Tasche September 15, 2003 Abstract Even in the simple Vasicek one-factor credit portfolio model, the exact contributions

More information

Regularly Varying Asymptotics for Tail Risk

Regularly Varying Asymptotics for Tail Risk Regularly Varying Asymptotics for Tail Risk Haijun Li Department of Mathematics Washington State University Humboldt Univ-Berlin Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin

More information

The largest eigenvalues of the sample covariance matrix. in the heavy-tail case

The largest eigenvalues of the sample covariance matrix. in the heavy-tail case The largest eigenvalues of the sample covariance matrix 1 in the heavy-tail case Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia NY), Johannes Heiny (Aarhus University)

More information

On Backtesting Risk Measurement Models

On Backtesting Risk Measurement Models On Backtesting Risk Measurement Models Hideatsu Tsukahara Department of Economics, Seijo University e-mail address: tsukahar@seijo.ac.jp 1 Introduction In general, the purpose of backtesting is twofold:

More information

Non-parametric Estimation of Elliptical Copulae With Application to Credit Risk

Non-parametric Estimation of Elliptical Copulae With Application to Credit Risk Non-parametric Estimation of Elliptical Copulae With Application to Credit Risk Krassimir Kostadinov Abstract This paper develops a method for statistical estimation of the dependence structure of financial

More information

Multivariate Heavy Tails, Asymptotic Independence and Beyond

Multivariate Heavy Tails, Asymptotic Independence and Beyond Multivariate Heavy Tails, endence and Beyond Sidney Resnick School of Operations Research and Industrial Engineering Rhodes Hall Cornell University Ithaca NY 14853 USA http://www.orie.cornell.edu/ sid

More information

Copulas for Markovian dependence

Copulas for Markovian dependence Mathematical Statistics Stockholm University Copulas for Markovian dependence Andreas N. Lagerås Research Report 2008:14 ISSN 1650-0377 Postal address: Mathematical Statistics Dept. of Mathematics Stockholm

More information

Tail process and its role in limit theorems Bojan Basrak, University of Zagreb

Tail process and its role in limit theorems Bojan Basrak, University of Zagreb Tail process and its role in limit theorems Bojan Basrak, University of Zagreb The Fields Institute Toronto, May 2016 based on the joint work (in progress) with Philippe Soulier, Azra Tafro, Hrvoje Planinić

More information

arxiv: v1 [q-fin.rm] 12 Mar 2013

arxiv: v1 [q-fin.rm] 12 Mar 2013 Understanding Operational Risk Capital Approximations: First and Second Orders Gareth W. Peters 1,2,3 Rodrigo S. Targino 1 Pavel V. Shevchenko 2 arxiv:1303.2910v1 [q-fin.rm] 12 Mar 2013 1 Department of

More information

Stochastic volatility models: tails and memory

Stochastic volatility models: tails and memory : tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier Plan Model assumptions; Limit theorems for partial sums and

More information

Copulas for Markovian dependence

Copulas for Markovian dependence Bernoulli 16(2), 2010, 331 342 DOI: 10.3150/09-BEJ214 Copulas for Markovian dependence ANDREAS N. LAGERÅS Department of Mathematics, Stockholm University, SE-10691 Stockholm, Sweden. E-mail: andreas@math.su.se

More information

arxiv: v2 [math.pr] 23 Jun 2014

arxiv: v2 [math.pr] 23 Jun 2014 COMPUTATION OF COPULAS BY FOURIER METHODS ANTONIS PAPAPANTOLEON arxiv:08.26v2 [math.pr] 23 Jun 204 Abstract. We provide an integral representation for the (implied) copulas of dependent random variables

More information

Multivariate Markov-switching ARMA processes with regularly varying noise

Multivariate Markov-switching ARMA processes with regularly varying noise Multivariate Markov-switching ARMA processes with regularly varying noise Robert Stelzer 23rd June 2006 Abstract The tail behaviour of stationary R d -valued Markov-Switching ARMA processes driven by a

More information

Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns

Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns Ruin Probability for Non-standard Poisson Risk Model with Stochastic Returns Tao Jiang Abstract This paper investigates the finite time ruin probability in non-homogeneous Poisson risk model, conditional

More information

Lévy processes and Lévy copulas with an application in insurance

Lévy processes and Lévy copulas with an application in insurance Lévy processes and Lévy copulas with an application in insurance Martin Hunting Thesis for the degree of Master of Science Mathematical statistics University of Bergen, Norway June, 2007 This thesis is

More information

A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS

A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS Statistica Sinica 20 2010, 365-378 A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS Liang Peng Georgia Institute of Technology Abstract: Estimating tail dependence functions is important for applications

More information

Empirical Examination of Operational Loss Distributions

Empirical Examination of Operational Loss Distributions Empirical Examination of Operational Loss Distributions Svetlozar T. Rachev Institut für Statistik und Mathematische Wirtschaftstheorie Universität Karlsruhe and Department of Statistics and Applied Probability

More information

Optimal Reinsurance Strategy with Bivariate Pareto Risks

Optimal Reinsurance Strategy with Bivariate Pareto Risks University of Wisconsin Milwaukee UWM Digital Commons Theses and Dissertations May 2014 Optimal Reinsurance Strategy with Bivariate Pareto Risks Evelyn Susanne Gaus University of Wisconsin-Milwaukee Follow

More information

Analysis methods of heavy-tailed data

Analysis methods of heavy-tailed data Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia February, 13-18, 2006, Bamberg, Germany June, 19-23, 2006, Brest, France May, 14-19, 2007, Trondheim, Norway PhD course Chapter

More information

Exponential Distribution and Poisson Process

Exponential Distribution and Poisson Process Exponential Distribution and Poisson Process Stochastic Processes - Lecture Notes Fatih Cavdur to accompany Introduction to Probability Models by Sheldon M. Ross Fall 215 Outline Introduction Exponential

More information

A Note on Tail Behaviour of Distributions. the max domain of attraction of the Frechét / Weibull law under power normalization

A Note on Tail Behaviour of Distributions. the max domain of attraction of the Frechét / Weibull law under power normalization ProbStat Forum, Volume 03, January 2010, Pages 01-10 ISSN 0974-3235 A Note on Tail Behaviour of Distributions in the Max Domain of Attraction of the Frechét/ Weibull Law under Power Normalization S.Ravi

More information

Copulas and dependence measurement

Copulas and dependence measurement Copulas and dependence measurement Thorsten Schmidt. Chemnitz University of Technology, Mathematical Institute, Reichenhainer Str. 41, Chemnitz. thorsten.schmidt@mathematik.tu-chemnitz.de Keywords: copulas,

More information

Non parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015.

Non parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015. Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, 2015. a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire

More information

Multivariate Pareto distributions: properties and examples

Multivariate Pareto distributions: properties and examples Multivariate Pareto distributions: properties and examples Ana Ferreira 1, Laurens de Haan 2 1 ISA UTL and CEAUL, Portugal 2 Erasmus Univ Rotterdam and CEAUL EVT2013 Vimeiro, September 8 11 Univariate

More information

Approximation and Aggregation of Risks by Variants of Panjer s Recursion

Approximation and Aggregation of Risks by Variants of Panjer s Recursion Approximation and Aggregation of Risks by Variants of Panjer s Recursion Uwe Schmock Vienna University of Technology, Austria Version of Slides: Feb. 8, 2013 Presentation based on joint work with Stefan

More information

Theoretical Sensitivity Analysis for Quantitative Operational Risk Management

Theoretical Sensitivity Analysis for Quantitative Operational Risk Management Theoretical Sensitivity Analysis for Quantitative Operational Risk Management Takashi Kato arxiv:1104.0359v5 [q-fin.rm] 24 May 2017 First Version: April 3, 2011 This Version: May 24, 2017 Abstract We study

More information

Chapter 5. Chapter 5 sections

Chapter 5. Chapter 5 sections 1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions

More information

STOCHASTIC GEOMETRY BIOIMAGING

STOCHASTIC GEOMETRY BIOIMAGING CENTRE FOR STOCHASTIC GEOMETRY AND ADVANCED BIOIMAGING 2018 www.csgb.dk RESEARCH REPORT Anders Rønn-Nielsen and Eva B. Vedel Jensen Central limit theorem for mean and variogram estimators in Lévy based

More information

A Brief Introduction to Copulas

A Brief Introduction to Copulas A Brief Introduction to Copulas Speaker: Hua, Lei February 24, 2009 Department of Statistics University of British Columbia Outline Introduction Definition Properties Archimedean Copulas Constructing Copulas

More information

Modelling Operational Risk Using Bayesian Inference

Modelling Operational Risk Using Bayesian Inference Pavel V. Shevchenko Modelling Operational Risk Using Bayesian Inference 4y Springer 1 Operational Risk and Basel II 1 1.1 Introduction to Operational Risk 1 1.2 Defining Operational Risk 4 1.3 Basel II

More information

Estimation of Operational Risk Capital Charge under Parameter Uncertainty

Estimation of Operational Risk Capital Charge under Parameter Uncertainty Estimation of Operational Risk Capital Charge under Parameter Uncertainty Pavel V. Shevchenko Principal Research Scientist, CSIRO Mathematical and Information Sciences, Sydney, Locked Bag 17, North Ryde,

More information

Multivariate Generalized Ornstein-Uhlenbeck Processes

Multivariate Generalized Ornstein-Uhlenbeck Processes Multivariate Generalized Ornstein-Uhlenbeck Processes Anita Behme TU München Alexander Lindner TU Braunschweig 7th International Conference on Lévy Processes: Theory and Applications Wroclaw, July 15 19,

More information