BSDEs with terminal conditions that have bounded Malliavin derivative

Abstract

We show existence and uniqueness of solutions to BSDEs of the form $$Y_t=\xi +{\int }_t^{T}f(s,Y_s,Z_s)ds-{\int }_t^T{Z}_{s}d{W}_s$$ in the case where the terminal condition $\xi$ has bounded Malliavin derivative. The driver $f(s,y,z)$ is assumed to be Lipschitz continuous in $y$ but only locally Lipschitz continuous in $z$. In particular, it can grow arbitrarily fast in $z$. If in addition to having bounded Malliavin derivative, $\xi$ is bounded, the driver needs only be locally Lipschitz continuous in $y$. In the special case where the BSDE is Markovian, we obtain existence and uniqueness results for semilinear parabolic PDEs with non-Lipschitz nonlinearities. We discuss the case where there is no lateral boundary as well as lateral boundary conditions of Dirichlet and Neumann type.