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

Summary This thesis deals with numerical methods for degenerate nonlinear partial differential equations (PDEs), as well as for control problems of nonlinear PDEs resulting from a new optimal transport problem. All these questions are motivated by applications in financial mathematics. The thesis is divided into four parts. In the first part, we focus on the necessary and sufficient condition of monotonicity of the finite difference theta-schema for the diffusion equation in dimension one. We give the explicit formula in the case of the heat equation, which is weaker than the classical Courant-Friedrichs-Lewy (CFL) condition. In a second part, we consider a degenerate nonlinear parabolic PDE and propose a splitting scheme to solve it. This scheme combines a probabilistic scheme and a semi-Lagrangian scheme. Finally, it can be considered as a Monte-Carlo scheme. We give a convergence result and also a convergence rate of the scheme. In a third part, we study an optimal transport problem, where the mass is transported by a controlled drift-diffusion state process. The associated cost depends on the trajectories of the state process, its drift and its diffusion coefficient. The transport problem consists in minimizing the cost among all dynamics verifying the initial and terminal constraints on the marginal distributions. We prove a duality formulation for this transport problem, thus extending Kantorovich's duality to our context. The dual formulation maximizes a value function on the space of bounded continuous functions, and the corresponding value function for each bounded continuous function is the solution of an optimal stochastic control problem. In the Markovian case, we prove a dynamic programming principle for these optimal control problems, propose a projected gradient algorithm for the numerical solution of the dual problem, and prove its convergence. Finally, in a fourth part, we further develop the dual approach for the optimal transportation problem with an application to the search for arbitrage-free price bounds of variance options given European option prices. After a first analytical approximation, we propose a projected gradient algorithm to approximate the bound and the corresponding static strategy in vanilla options.