Processus de Lévy en Finance - Laboratoire de Probabilités et ...

188 CONCLUSIONS AND PERSPECTIVES
general framework in which the dependence structures of multidimensional Lévy processes can
be described. Lévy copulas completely characterize the possible dependence patterns of Lévy
processes in the sense that for every Lévy process, there exists a Lévy copula that describes its
dependence structure and for every Lévy copula and every n one-dimensional Lévy processes,
there exists an n-dimensional Lévy process with this Lévy copula and with margins given by
these one-dimensional processes. Multidimensional Lévy process models for applications can
thus be constructed by taking any n one-dimensional processes and a Lévy copula from a
(possibly parametric) family. The simulation methods, developed in the last chapter of this
thesis, allow to compute various quantities of interest in a Lévy copula model using the Monte
Carlo method.
The scope of potential applications of Lévy copula models in finance and other domains is
large. Financial applications include basket option pricing, and portfolio management. Lévy
copula models are also useful in insurance and in risk management, to model the dependence
between loss processes of different business lines, and more generally, in all multivariate problems
where dependence between jumps needs to be taken into account.
From the point of view of applications, the next step is to develop the methods of estimating
Lévy copula models from the data, using, for example, simulation-based techniques of statistical
inference [51]. A more theoretical research direction that we currently pursue in collaboration
with Jan Kallsen is to investigate the relation between the Lévy copula of a Lévy process and
its probabilistic copula at a given time.