L2-time regularity of BSDEs with irregular terminal functions

Emmanuel Gobet 1 Azmi Makhlouf 1
1 MATHFI - Mathématiques financières
LJK - Laboratoire Jean Kuntzmann
Abstract : 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.
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Stochastic Processes and their Applications, Elsevier, 2010, 120 (7), pp.1105-1132. 〈10.1016/j.spa.2010.03.003〉
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Emmanuel Gobet, Azmi Makhlouf. L2-time regularity of BSDEs with irregular terminal functions. Stochastic Processes and their Applications, Elsevier, 2010, 120 (7), pp.1105-1132. 〈10.1016/j.spa.2010.03.003〉. 〈hal-00291768〉

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