\(\mathbf L_2\)-time regularity of BSDEs with irregular terminal functions. (English) Zbl 1195.60079
Summary: We study the \({\mathbf L}_2\)-time regularity of the \(Z\)-component of a Markovian BSDE, whose terminal condition is a function \(g\) of a forward SDE \((X_t)_{0\leq t\leq T}\). When \(g\) is Lipschitz continuous, J. Zhang [Ann. Appl. Probab. 14, No. 1, 459–488 (2004(2004; Zbl 1056.60067)] proved that the related squared \({\mathbf L}_2\)-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.
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
Keywords:
backward stochastic differential equations; time regularity; Malliavin calculus; rate of convergenceCitations:
Zbl 1056.60067References:
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