On two numerical problems in applied probability : discretization of Stochastic Differential Equations and optimization of an expectation depending on a parameter.

Summary In the present paper, we first deal with the discretization of stochastic dierential equa- tions. We elaborate on the analysis of the weak error of the Euler scheme by Talay and Tubaro (31) to contruct schemes with quicker weak rate of convergence for SDEs corresponding to an infinitesimal generator with smooth coecients. We also extend this analysis to the case of a discontinuous drift coecient. In a second part, we present two applications of stochastic gradient algorithms in finance.