Stochastic control on networks.

Summary This thesis is divided into three main parts, dealing with quasi-linear parabolic PDEs on a junction, stochastic diffusions on a junction, and optimal control also on a junction, with control at the junction point. We start in the first chapter by introducing a new class of non-degenerate quasi-linear PDEs, satisfying a non-linear and non-dynamic Neumann (or Kirchoff) condition at the junction point. We prove the existence of a classical solution, as well as its uniqueness. One of the motivations for studying this type of PDE is to make the link with the theory of optimal control on junctions, and to characterize the value function of this type of problem using the Hamilton Jacobi Bellman equations. Thus, in the next chapter, we formulate a proof giving the existence of a diffusion on a junction. This process admits a local time, whose existence and quadratic variation depend essentially on the assumption of ellipticity of the second order terms at the junction. We will formulate an Itô formula for this process. Thus, thanks to the results of these two Chapters, we will formulate in the last Chapter a stochastic control problem on junctions, with control at the junction point. The control space is that of probability measures solving a martingale problem. We prove the compactness of the space of admissible controls, as well as the principle of dynamic programming.