My domains of research are optimal control (finite and infinite dimensional), static and dynamic optimization, partial differential equations (finite and infinite dimensional)-linked to optimal control problems-,Mean Field Games and applications to economics and mathematical finance. Below I describe my fields of research in details.
Homogenization and singular perturbations problems for HJ equations, with applications to large deviations of multiscale stochastic volatility models and pricing of out-of-the-money options near maturity; integro-partial differential equations of HJ type related to Lévy processes; finite horizon optimal control problems with discontinuous dynamics and costs.
Mean Field Games (MFGs) with applications to economics:
MFGs with strong aggregation (strong decreasing cost) for aggregation phenomena.
MFGs models in economics for the time evolution of the human capital, in particular with spatial interactions in the human capital. These MFGs share a nonstandard challenging structure not covered by the existing literature: non monotone (in the distribution) cost and non separable (in the gradient of the value function and the distribution) Hamiltonian. Epidemiological models in such MFGs economic models where the rate of infection depends on the distribution of the population with the goal of modeling human capital accumulation during the spread of a disease following an agent based approach, where agents behave maximizing their intertemporal utility.
Mean field games and mean field control for dynamical models for the environmental quality. We study the evolution of the quality of the environment in economies with a large number of species where the goal is to increase the global quality of the environment and at the same time maximize the gain. We study two different scenarios: we search first for non cooperative Nash equilibria and then we consider the situation in which agents cooperate and there is a social planner.
Mean Field Games in infinite dimension with applications. In the class of problems we study the state equation is a PDE or a delay equation which can be written as an ODE in a suitable Hilbert space. The applications we are interested in are a vintage capital model where the state equation for the capital is a first order PDE and to a model for a production output planning problem with delay in the control variable and to a vintage capital model. We study the Mean Field Game (MFG) system and the Master equation (ME), and we prove convergence of the discrete model (MFG) to the continuous one (ME). We also prove a verification theorem.
Infinite dimensional Hamilton-Jacobi equations, arising in a Mean Field type control problem in for a Ramsey-type economic model.
Optimal control and optimization in economics:
Habits and demand changes after COVID 19 (i.e. pandemic economy with lockdown of a sector of the economy). The goal is to show that the demand of the goods produced by the sector closed during the lockdown could shrink or expand with respect to their pre-pandemic level depending on the lockdown’s duration and the habits’ strength. Moreover, the end of a lockdown may be characterized by a price surge due to a combination of strong demand of both goods and rigidities in production
Sparse optimization:
Non convex non smooth sparse optimization, analytical and numerical aspects, with applications in optimal control, reconstruction of images, and fractures mechanics.
Inverse problems in fractures mechanics for the identification of crack by shape optimization techniques.
In my bachelor and master thesis I worked in calculus of variations and optimization. Within this area of research my specific interests have been isoperimetric and Brunn-Minkowski inequalities for variational functionals and quantitative results.