Approximation and density estimation for stochastic evolution equations.

Summary In the first part of this thesis, we obtain the existence of a density and Gaussian estimates for the solution of a backward stochastic differential equation. This is an application of Malliavin's calculus and more particularly of a formula of I. Nourdin and F. Viens. The second part of this thesis is devoted to the simulation of a stochastic partial differential equation by a probabilistic method based on the representation of the stochastic partial differential equation in terms of a backward stochastic differential equation, introduced by E. Pardoux and S. Peng. We extend in this framework the ideas of F. Zhang and E. Gobet et al. on the simulation of a backward stochastic differential equation. In the last part, we study the weak error of the implicit Euler scheme for diffusion processes and the stochastic heat equation. In the first case, we extend the results of D. Talay and L. Tubaro. In the second case, we extend the work of A. Debussche.