Improved Convergence Rate for Reflected BSDEs by Penalization Method

We investigate the convergence of numerical solution of Reflected Backward Stochastic Differential Equations (RBSDEs) using the penalization approach in a general non-Markovian framework. We prove the convergence between the continuous penalized solution and the reflected one, in full generality, at order 1/2 as a function of the penalty parameter; the convergence order becomes 1 when the increasing process of the RBSDE has a bounded density, which is a mild condition in practice. The convergence is analyzed in a.s.-sense and -sense (). To achieve these new results, we have developed a refined analysis of the behavior of the process close to the barrier. Then we propose an implicit scheme for computing the discrete solution of the penalized equation and we derive that the global convergence order is 3/8 as a function of time discretization under mild regularity assumptions. This convergence rate is verified in the case of American put options and some numerical tests illustrate these results.