Speaker
Description
In the traditional approach, as well as in the one nowadays usually referred to as Lagrangian, the Hamilton–Jacobi equation, with Hamiltonian given by the convex dual of the Lagrangian cost, plays a crucial role in the analysis of first-order MFG problems. The main reason is that the vector field driving the continuity equation is expressed in terms of Hp(·, ·, −Dv(x, t)), where v is the value function. To make this expression meaningful, some analysis of the HJ equation is required, typically to establish semiconcavity properties of v and, in turn, enough differentiability along the relevant characteristics.
On the contrary, we show that a vector field driving the continuity equation can be defined under rather general assumptions on the Lagrangian, solely by exploiting the rep- resentation formula for the value function, without ever using the fact that it solves the corresponding HJ equation. This paves the way for a purely variational analysis of the problem and, in particular, allows us to dispense with the separable structure commonly imposed on Lagrangians in the MFG literature.
This is ongoing research in collaboration with Marco Pozza (Unilink).