Stochastic control methods for optimal transportation and probabilistic numerical schemes for PDEs.

Summary This thesis deals with the numerical methods for a fully nonlinear degenerate parabolic partial differential equations (PDEs), and for a controlled nonlinear PDEs problem which results from a mass transportation problem. The manuscript is divided into four parts. In a first part of the thesis, we are interested in the necessary and sufficient condition of the monotonicity of finite difference $\theta$-scheme for a one-dimensional diffusion equations. An explicit formula is given in case of the heat equation, which is weaker than the classical Courant-Friedrichs-Lewy (CFL) condition. In a second part, we consider a fully nonlinear degenerate parabolic PDE and propose a splitting scheme for its numerical resolution. The splitting scheme combines a probabilistic scheme and the semi-Lagrangian scheme, and in total, it can be viewed as a Monte-Carlo scheme for PDEs. We provide a convergence result as well as a rate of convergence. In the third part of the thesis, we study an optimal mass transportation problem. The mass is transported by the controlled drift-diffusion dynamics, and the associated cost depends on the trajectories, the drift as well as the diffusion coefficient of the dynamics. We prove a strong duality result for the transportation problem, thus extending the Kantorovich duality to our context. The dual formulation maximizes a value function on the space of all bounded continuous functions, and every value function corresponding to a bounded continuous function is the solution to a stochastic control problem. In the Markovian cases, we prove the dynamic programming principle of the optimal control problems, and we propose a gradient-projection algorithm for the numerical resolution of the dual problem, and provide a convergence result. Finally, in a fourth part, we continue to develop the dual approach of mass transportation problem with its applications in the computation of the model-independent no-arbitrage price bound of the variance option in a vanilla-liquid market. After a first analytic approximation, we propose a gradient-projection algorithm to approximate the bound as well as the corresponding static strategy in vanilla options.