Linee di ricerca
Various classes of optimization problems and games for stochastic processes are studied: optimal control, optimal stopping, impulse or singular control, stochastic differential games, and mean-field games. The models encompass stochastic differential equations, particle systems with weak interaction, McKean-Vlasov equations, point processes, and stochastic partial differential equations. The techniques span backward stochastic differential equations, the stochastic maximum principle and Hamilton-Jacobi-Bellman equations (generally understood in a viscosity sense and possibly on measure spaces or other specific spaces). Both theoretical and applied aspects (economics, finance, energy, climate risk, …) of these models are considered.
Faculty
Luciano Campi
Andrea Cosso
Marco Fuhrman
PhDs and post-docs
Federico Cannerozzi
Silvia Rudà
Laura Perelli
Antonio Scognamiglio
The research line focuses on different aspect of dynamical stochastic processes, from the theoretical, computational and applicative point of view. In particular:
1. Interacting particles systems and multi-scales problems: mathematical modelling at the micro or nano-scale of the individual level and right rescaling via laws of large numbers and mean-field theory.
2. Modelling and Statistics of general Levy processes: we consider different applications from energy markets to evolution of pollutants, with particular interest to the parameter estimation of the SDE driven by general Levy processes, starting by observed times series.
3. Symmetries and invariants in stochastic dynamics: generalization of the classical S. Lie symmetries approach to ODE and PDE to the stochastic case (SDE). The research involves new definitions of the concepts of symmetries and invariance properties.
4. A stochastic approach to Bose-Einstein Condensation: a stochastic approach to Bose-Einstein condensation phenomenon and to the involved equations is developed within Nelson stochastic mechanics via interacting diffusive particle techniques.
Faculty
Daniela Morale
Stefania Ugolini
Federico Sau
PhDs and Post-docs
Susanna Dehç
Ernesto Maria Greco
Giulia Rui
The main interests of this research topic are the following: theory and statistics of spatial point processes and random sets in Euclidean spaces, with a special focus on to the study and the estimation of the intensity and of the mean volume and boundary densities; birth-and-growth processes and dynamic germ-grain models for the study of phase transition processes in crystallization processes in materials sciences. Techniques used include elements of geometric measure theory and large deviation theory for both probabilistic and statistical aspects.
Faculty
This research line includes the well-posedness and both analytical and numerical regularity of solutions of PDE not strictly parabolic with stochastic dynamical boundary conditions. This means from one side to study the regularity properties in the Sobolev spaces of the solution of PDE equations with a stochastic boundary condition, which is only Hölder continuous; on the other side it means to develop numerical methods for an efficient and accurate approximation of the solution. This problem arises in the study of degradation phenomena in cultural heritage due to environmental pollutants and climate changes.
In this research line we also focus on the investigation of stochastic control problems in infinite dimension with state process satisfying a stochastic PDE. Finally, we are interested in the problem of stochastic filtering and the study of the so-called Zakai equation.
Faculty
Andrea Cosso
Marco Fuhrman
Daniela Morale
Federico Sau
Stefania Ugolini
Persone
