-time regularity of BSDEs with irregular terminal functions

We study the -time regularity of the Z-component of a Markovian BSDE, whose terminal condition is a function g of a forward SDE (Xt)0<=t<=T. When g is Lipschitz continuous, Zhang (2004) [18] proved that the related squared -time regularity is of order one with respect to the size of the time mesh. We extend this type of result to any function g, including irregular functions such as indicator functions for instance. We show that the order of convergence is explicitly connected to the rate of decreasing of the expected conditional variance of g(XT) given Xt as t goes to T. This holds true for any Lipschitz continuous generator. The results are optimal.