Supervisor: Matthias Winkel Prerequisites: BS3a Applied Probability

Type: Theoretical project with simulation component if desired Description: Copulas describe the dependence structure of multivariate distributions (or random variables), without reference to the marginal distributions. More formally, copulas are multivariate distribution functions of random variables (U1,..., Un) on [0, 1] n, for which each Ui is uniformly distributed. Sklar’s Theorem states that a (continuous) multivariate distribution function H can be uniquely expressed in terms of its marginal distributions and a copula. The project could start by working out a detailed proof of this theorem. In practice, one needs specific families of copulas just as one works with specific families of distributions and there is a multitude of methods to construct copulas. The project will include the exposition of some of these. There are also several summary statistics of dependence structure such as the well-known correlation coefficient. In the context of copulas, the most important ones are Kendall’s τ and Spearman’s ρ. Copulas have become very important in finance and insurance over the last few years