Currently, my research is mainly about macroeconomic models with heterogeneous agents, where I use strongly the theory of stochastic control and mean field games. These models have been designed to explain how the vast heterogeneity at the micro level (mainly in income and wealth) determines the optimal choices of the individual agents and the shape of the economic aggregates, and how the optimal choices of the agents under some macroeconomic policies affect the distribution of the economic variables across the time, including the distribution of income and wealth. For those interested in this topic, I strongly recommend the research and the courses of Benjamin Moll.
Although their importance to answer some of the most important questions in economics, equilibria in macroeconomic models with heterogeneous agents are usually characterized by couple systems of nonlinear partial differential equations, which are hard to solve, so these models can be interesting for well experimented mathematicians.
During my PhD, I hope to contribute to some currently open mathematical problems about this topic, discussed in Partial differential equation models in macroeconomics, written by Achdou, Y. et al. in 2014. Particularly, I will work on the following problems in macroeconomic models with heterogeneous agents:
Existence and uniqueness of equilibria
New methods to calibrate this class of models
Regularization methods to approximate equilibria in macroeconomic models with heterogeneous agents and common noise using mean field games models without common noise
Other topics of interest for academic research include statistics and health economics.