Population dynamics: stochastic control and hybrid modeling of cancer.

Summary The objective of this thesis is to develop the theory of stochastic control and its applications in population dynamics. From a theoretical point of view, we present the study of stochastic control problems with finite horizon on diffusion, nonlinear branching and branch-diffusion processes. In each case, we reason by the dynamic programming method, taking care to carefully prove a conditioning argument analogous to the strong Markov property for controlled processes. The principle of dynamic programming then allows us to prove that the value function is a solution (regular or viscosity) of the corresponding Hamilton-Jacobi-Bellman equation. In the regular case, we also identify a Markovian optimal control by a verification theorem. From an application point of view, we are interested in the mathematical modeling of cancer and its therapeutic strategies. More precisely, we build a hybrid model of tumor growth that accounts for the fundamental role of acidity in the evolution of the disease. The targets of therapy are explicitly included as parameters of the model in order to use it as a support for the evaluation of therapeutic strategies.