BSDEs with terminal conditions that have bounded Malliavin derivative

We show existence and uniqueness of solutions to BSDEs of the formYt=ξ+∫tTf(s,Ys,Zs)ds−∫tTZsdWs in the case where the terminal condition ξ   has bounded Malliavin derivative. The driver f(s,y,z)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, ξ 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.