*Tid:***7 februari 2000 kl 1515-1700 **

*Plats :***Seminarierummet 3733**, Institutionen för
matematik, KTH, Lindstedts väg 25, plan 7. Karta!

*Föredragshållare:***
Filip Lindskog
**

**Titel:** **
Modelling Dependence with Copulas
**

*Sammanfattning: *

The dependence between random variables is completely described by their joint distribution. However, dependence and marginal behaviour can be separated. The copula of a multivariate distribution can be considered to be the part describing the dependence structure. Furthermore, strictly increasing transformations of the underlying random variables result in the transformed variables having the same copula. Hence copulas are invariant under strictly increasing transformations of the margins. This provides a way of studying scale-invariant measures of associations and also a starting point for construction of multivariate distributions. Scale-invariant measures of association such as Kendall's tau and Spearman's rho only depend on the copula and are thus invariant under strictly increasing transformations of the margins, which means that we can apply arbitrary continuous margins to our chosen copula leaving among other things the measures of association unchanged.

Tail dependence and Kendall's tau and Spearman's rho are presented and evaluated for a large number of copula families. Among these copula families are families suitable for modelling extreme events, which are highly relevant as a basis for risk models in insurance and finance.

The multivariate normal distribution and linear correlation are the basis of most models used to model dependence. Even though this distribution has a wide range of dependence it is quite seldom suitable for modelling real world situations in insurance and finance. We will show that using a model based on the multivariate normal distribution without knowledge of its limitations can prove very dangerous. Linear correlation is a natural measure of dependence in the context of the normal distribution. However, it should be noted that it is not invariant under strictly increasing transformations of the marginals and can be misleading as a measure of dependence.

The problem of simulating dependent data arises naturally in Monte Carlo approaches to risk management. One main aim of this paper is to show that when addressing this problem knowledge of copulas and copula based dependence concepts is important, and also the usefulness of copula ideas in this approach to risk management Another main aim of this paper is the construction of multivariate extensions of bivariate copula families. In particular we focus on multivariate extensions with a flexible and wide range of dependence for which efficient algorithms for random variate generation are presented.