Numerical methods and deep learning for stochastic control problems and partial differential equations.

Summary The thesis deals with numerical schemes for Markovian decision problems (MDPs), partial differential equations (PDEs), backward stochastic differential equations (SRs), as well as reflected backward stochastic differential equations (SRDEs). The thesis is divided into three parts.The first part deals with numerical methods for solving MDPs, based on quantization and local or global regression. A market-making problem is proposed: it is solved theoretically by rewriting it as an MDP. and numerically by using the new algorithm. In a second step, a Markovian embedding method is proposed to reduce McKean-Vlasov type probabilities with partial information to MDPs. This method is implemented on three different McKean-Vlasov type problems with partial information, which are then numerically solved using numerical methods based on regression and quantization.In the second part, new algorithms are proposed to solve MDPs in high dimension. The latter are based on neural networks, which have proven in practice to be the best for learning high dimensional functions. The consistency of the proposed algorithms is proved, and they are tested on many stochastic control problems, which allows to illustrate their performances.In the third part, we focus on methods based on neural networks to solve PDEs, EDSRs and reflected EDSRs. The convergence of the proposed algorithms is proved and they are compared to other recent algorithms of the literature on some examples, which allows to illustrate their very good performances.