Probabilistic interpretations of operators in divergence form and analysis of associated probabilistic numerical methods.

Summary The analysis and approximation of solutions of stochastic differential equations (SDEs) with discontinuous coefficients is a subject that has not been treated in a fully satisfactory way. This problem becomes particularly motivating when one tries to approximate, by Monte-Carlo methods, the solutions of some partial differential equations (PDE) which also involve discontinuous coefficients. This is for example the case, well known in physics, of PDEs with divergence operator (DIO) whose coefficients are discontinuous and which we study in this thesis: the discontinuities then translate the irregularities of the medium in which the system studied evolves. This thesis proposes new results for the analysis and approximation of solutions of E. D. S. which are related to an O. F. D. whose coefficients are discontinuous. The statistical aspects of the models involved are also studied.