Relaxed formulation for Controlled Branching Diffusions and Scaling Limits

par Antonio Ocello

mardi 5 nov. 2024, 14:00 → 15:00 Europe/Paris

In this presentation, we delve into optimal control problems for branching diffusion processes, beginning with a detailed formulation of the associated HJB equation and a verification theorem for this class of processes. We demonstrate that, under symmetry constraints, the optimal control can be confined to a subset of control processes that respect these symmetry conditions. This reduction allows us to reformulate the problem through an equivalent, relaxed approach based on martingale measures, which offers a novel perspective and useful characterizations. We introduce the concept of atomic control and prove its equivalence to strong controls within this relaxed framework, as well as its equivalence to the original strong problem, under a Filippov-type convexity condition.   
Using the previous formalism, we explore one scaling limit for these processes that ensures the existence of a new class of controlled superprocesses. Focusing on this new class of processes and by leveraging the density of cylindrical functions in the space of continuous functions on finite measures, we associate this new optimization problem with an HJB equation  on the space of finite measures and establish a verification theorem. Finally, focusing on an exponential-type value function, we show that a regular solution to a finite-dimensional nonlinear PDE can be used to provide a smooth solution to the previously equation, in this highly symmetric case. This talk is based on recent research presented in arXiv:2304.07064 and arXiv:2306.15962.