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.
From a theoretical point of view, as emphasized in the previous sections, the main novelty of our project is given by the use of multivariate inverse subordinators, whose components are not independent. This permits us to take into account a more complicated correlation structure in the time-changed processes. Correspondingly, the pseudodifferential operators involved are anisotropic counterparts of the well-known fractional Laplacian operator. Thus many theoretical results on the latter, which have been extensively treated in the literature, could be adapted, with the necessary modifications, to the present case.
On the other hand, from the point of view of the applications, very interesting special cases could also been obtained from the general theory: in particular, when the leading process is a multivariate Brownian motion or a multivariate Poisson process, many results can be applied, in mathematical finance, to multivariate pricing and risk evaluation.
Finally, in this context, the two aspects of parameter estimation and simulation techniques are completely missing in the literature; therefore we are aimed at filling this gap.
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. The latter can be applied in this context to derive moments and limiting distributions (in the large box limit) of geometrical quantities like length of level curves, excursion area, length of nodal lines, number of critical points, Euler characteristic of the excursion sets.
We will start our analysis focusing on the more tractable geometrical quantities (length of level curves and excursion area) in the case of bivariate subordinators with independent components. The case of bivariate subordinators without independent components is more challenging. In fact, in the anisotropic case, the geometrical functionals under consideration are not constant over the space and this requires a more careful analysis and heavier calculations. For this reason in the anisotropic case we will first focus on the derivation of the first moment of the (random) geometrical functionals.