L2-time regularity of BSDEs with irregular terminal functions

We study the L 2 -time regularity of the Z -component of a Markovian BSDE, whose terminal condition is a function g of a forward SDE ( X t ) 0 ≤ t ≤ T . When g is Lipschitz continuous, Zhang (2004) [18] proved that the related squared L 2 -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 ( X T ) given X t as t goes to T . This holds true for any Lipschitz continuous generator. The results are optimal.