My research focuses on both numerical and theoretical aspects of various types of stochastic differential equations, their related integro-differential operators of Kolmogorov type, and their applications to mathematical finance.
In particular, part of my activity has been concerned with the study of analytical approximations for the fundamental solutions to integro-differential (possibly non-linear) equations of Kolmogorov type, resulting in weak approximations of the related stochastic differential equations. Several features are allowed, including: Lévy-type jumps, degeneracies in the volatility coefficients, McKean-Vlasov-type (mean-field) interaction. The applications of these studies concern the fast valuation of financial derivatives, as well as the asymptotic properties of the related implied volatility surfaces in certain maturity/moneyness regimes.
In the hypoelliptic framework, I have worked on a class of diffusion processes satisfying a weak Hörmander-type condition. Such diffusions arise, among other fields of social and natural sciences, in mathematical finance, in particular in the valuation of arithmetic-Asian options. In particular, I have focused on the characterization of so-called intrinsic Holder spaces, as well as on their applications to the study of asymptotic and regularity properties of the transition densities in local settings.
I have also worked on some strong approximations for some classes of stochastic differential equations, such as backward SDEs with non-smooth driver, linear McKean-Vlasov jump-diffusions, and recently stochastic PDEs of parabolic type.
