Dynamic optimization of branching diffusion processes : Stochastic Control’s lens on particle systems and their scaling limits

Overview & Synopsis

The goal of this thesis is to uncover interesting structures occurring in the intersection of three distinct fields: stochastic control theory, branching diffusion processes, and McKean-Vlasov dynamics. In the ini- tial phase, we investigate potential extensions of the stochastic target problem and the optimal stopping problem within the context of branching processes. By constraining our examination to cost functions that respect the inherent symmetry of the problem, we show how the optimization of a global crite- rion can be recast as finite-dimensional optimization challenges through the utilization of a branching property. This finding paves the way to a differential characterization. Using a dynamic programming approach, we prove the value function is the unique viscosity solution to an HJB equation.The second part of this work delves into the theory of controlled branching diffusion processes, under a symmetrical structure in the cost function with respect to particle labeling. Exploring a relaxed formulation, we rewrite the control problem as the minimization of a lower semicontinuous function within a compact domain. This formulation, therefore, provides theoretical guarantees regarding the existence of a globally optimal solution. This abstract setting paves the way to scaling limits for these processes, leading to the class of controlled superprocesses. Within this dynamical framework, we establish an HJB equation in the space of finite measures. Moreover, for specific cost functions, we go back to the initial approach, retrieving regular solutions for the control problem through a branching property and finite-dimensional optimization.