Probabilistic numerical methods for solving transport equations and for evaluating exotic options.

Summary The aim of this thesis is the numerical analysis of probabilistic algorithms for the solution of transportation equations and for the computation of complex option prices in financial mathematics. In the part concerning the transport equations, we have constructed an algorithm for approximating the solution and we have obtained an estimate of the speed of convergence of the error as a function of the discretization step in time. We then validated this algorithm on test cases related to industrial problems. Our results are comparable to those provided by determinist methods used at the e. The financial mathematics part deals with the problem of approximating the expectation of functionals depending at most on a diffusion process. This problem is related to the evaluation of exotic options. We first give convergence speed results for particular functions. Then, in order to generalize these results for a large class of functions and diffusions, we study the regularity of the solution of a degenerate parabolic edp with neumann condition and we obtain accurate estimates of the derivatives.