Optimal control of branching diffusion processes: A finite horizon problem

Abstract

In this paper, we aim to develop the theory of optimal stochastic control for branching diffusion processes where both the movement and the reproduction of the particles depend on the control. More precisely, we study the problem of minimizing a criterion that is expressed as the expected value of the product of individual costs penalizing the final position of each particle. In this setting, we show that the value function is the unique viscosity solution of a nonlinear parabolic PDE, that is, the Hamilton-Jacobi-Bellman equation corresponding to the problem. To this end, we extend the dynamic programming approach initiated by Nisio [27] to deal with the lack of independence between the particles as well as between the reproduction and the movement of each particle. In particular, we exploit the particular form of the optimization criterion to recover a weak form of the branching property. In addition, we provide a precise formulation and a detailed justification of the adequate dynamic programming principle.

Mean field games with branching

“…where N n s := #K n s is the number of particles alive at time s. The proof of the proposition above can be found in Claisse [15,Proposition 2.1]. It relies essentially on two arguments which follow from Assumption 2.2.…”

“…The proposition above follows from Itô's formula, see [15,Proposition 3.2]. It is the starting point for the relaxed formulation which is introduced in Section 4.…”

“…where the supremum is taken over all controls satisfying (15). To conclude, it remains to see that the restriction of the cost function to such control is Lipchitz continuous in x, uniformly w.r.t.…”

“…In spite of its potential for applications, the optimal control of branching diffusion processes has not attracted much interest so far. We can mention nonetheless the papers of Ustunel [36], Nisio [30] and Claisse [15]. See also the related work of Bensoussan et al [5] on differential games.…”

Scaling limit for stochastic control problems in population dynamics

“…On the other side of the literature, Getz in [Get75] has studied control problems related to a birth/death process. This work has been extended more recently by Claisse in [Cla18] to branching processes. We refer to those models as discrete models.…”