L2-time regularity of BSDEs with irregular terminal functions

Emmanuel Gobet (1) , Azmi Makhlouf (1)

1 MATHFI - Mathématiques financières

We study the L2-time regularity of the $Z$-component of a Markovian BSDE, whose terminal condition is a function $g$ of a forward SDE $(X_t)_{0\le t\le T}$. When $g$ is Lipschitz continuous, Zhang '04 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 $g$, 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.

Mots clés: Backward stochastic differential equations; Time regularity; Malliavin calculus; Rate of convergence

Domaines: Probabilités [math.PR]

Emmanuel Gobet, Azmi Makhlouf. L2-time regularity of BSDEs with irregular terminal functions. Stochastic Processes and their Applications, 2010, 120 (7), pp.1105-1132. ⟨10.1016/j.spa.2010.03.003⟩. ⟨hal-00291768⟩