  In this work we study the numerical approximation of a class of ergodic
Backward Stochastic Differential Equations. These equations are formulated in
an infinite horizon framework and provide a probabilistic representation for
elliptic Partial Differential Equations of ergodic type. In order to build our
numerical scheme, we put forward a new representation of the PDE solution by
using a classical probabilistic representation of the gradient. Then, based on
this representation, we propose a fully implementable numerical scheme using a
Picard iteration procedure, a grid space discretization and a Monte-Carlo
approximation. Up to a limiting technical condition that guarantee the
contraction of the Picard procedure, we obtain an upper bound for the numerical
error. We also provide some numerical experiments that show the efficiency of
this approach for small dimensions.
