This talk is concerned with numerical methods for some non linear partial differential equations (PDEs) with obstacles arising in  stochastic optimal stopping time problems.

For linear PDEs, there are many high-order numerical schemes that have been analyzed in the literature.
However, for nonlinear PDEs with obstacles, it is very challenging to derive efficient second order schemes (in time and space), with rigorous stability estimates.

After reviewing some known results on finite difference methods,
we propose a new class of Backward Difference schemes  for obstacle problems and discuss some convergence and error estimate results. Numerical simulations will be presented on the american option  problem to illustrate the relevance of the numerical approach.