Ricerc@Sapienza

Multivariate subordinators are multidimensional Lévy processes that are increasing in each component , thus extending the property of the one-dimensional case. In the existing literature, the subordination of random vectors is limited to the case where the components are time-changed by independent subordinators.
Our aim is to start by studying the bivariate subordinators (with non-independent components) and the differential equation satisfied by their transition density. As particular case of our model we can recover the multivariate subordination of Brownian motions: indeed, in financial applications, different time changes for different assets, are more realistic and can be interpreted as a measure of the asset-specific trade.
The second step of our project can be the definition of the inverse (or hitting times) of the multivariate subordinators. In the one-dimensional case, the time-change of Lévy processes by means of inverse stable subordinators is currently used, for example, in order to generate anomalous diffusions.
Another possible direction of the research can be the study of geometrical properties of bivariate subordinators with or without independent components., by means of some probabilistic tools, such as Kac-Rice formula and Wiener chaos expansion.
Finally, parameters' estimation problems and simulation techniques could be implemented in this context, in order to integrate the theoretical work, from an application view point.