L2-time regularity of BSDEs with irregular terminal functions

We study the L2-time regularity of the ZZ-component of a Markovian BSDE, whose terminal condition is a function gg of a forward SDE (Xt)0≤t≤T(Xt)0≤t≤T. When gg is Lipschitz continuous, Zhang (2004) [18] proved that the related squared L2-time regularity is of order one with respect to the size of the time mesh. We extend this type of result to any function gg, 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)g(XT) given XtXt as tt goes to TT. This holds true for any Lipschitz continuous generator. The results are optimal.