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.

翻译：本文研究一类遍历型倒向随机微分方程的数值逼近问题。该类方程建立于无限时间框架下，并为遍历型椭圆偏微分方程提供了概率表示。为构建数值格式，我们通过使用梯度的经典概率表示，提出了偏微分方程解的一种新表示方法。基于此表示，我们结合Picard迭代过程、网格空间离散化与蒙特卡洛逼近，提出了一种完全可实现的数值格式。在保证Picard过程收敛性的极限技术条件下，我们得到了数值误差的上界估计。同时通过数值实验验证了该方法在低维情形下的有效性。