Algorithms for the resolution of stochastic control problems in high dimension by using probabilistic and max-plus methods.

Summary Finite-horizon stochastic optimal control problems are a class of optimal control problems involving stochastic processes considered over a bounded time interval. Like many optimal control problems, these problems are solved using the principle of dynamic programming which induces a partial differential equation (PDE) called Hamilton-Jacobi-Bellman equation. Methods based on the discretization of the space in the form of a grid, probabilistic methods or more recently max-plus methods can then be used to solve this equation. However, the first type of method fails when a high dimensional space is considered because of the curse of the dimension while the second type of method has so far only allowed to solve problems where the non-linearity of the partial differential equation with respect to the Hessian is not too strong. As for the third type of method, it leads to an explosion of the complexity of the value function. In this thesis, we introduce two new probabilistic schemes to enlarge the class of problems that can be solved by probabilistic methods. One is adapted to PDEs with bounded coefficients while the other can be applied to PDEs with bounded or unbounded coefficients. We prove the convergence of both probabilistic schemes and obtain estimates of the convergence error in the case of PDEs with bounded coefficients. We also give some results on the behavior of the second scheme in the case of PDEs with unbounded coefficients. Then, we introduce a completely new method for solving finite horizon stochastic optimal control problems which we call the max-plus probabilistic method. It allows to use the nonlinear character of max-plus methods in a probabilistic context while controlling the complexity of the value function. An application to the computation of the over-replication price of an option in an uncertain correlation model is given in the case of a 2 and 5 dimensional space.