The Maximum Principle for Optimal Control of BSDEs with Locally Lipschitz Coefficients

The present paper studies a stochastic control problem for a locally Lipschitz backward stochastic differential equation. Assuming that the control domain is not necessarily convex, we establish a necessary and sufficient condition for optimality satisfied by all optimal controls. These conditions are described by a linear locally Lipschitz SDE and a maximum condition on the Hamiltonian. We first prove, under some convenient conditions, the existence of a unique solution to the resulting adjoint equation. Then, with the help of an approximation argument on the coefficients, we define a family of control problems with globally Lipschitz coefficients whereby we derive a stochastic maximum principle for near optimality to such approximated systems. Thereafter, we turn back to the original control problem by passing to the limits. As far as the authors are aware, this is the first version of the stochastic maximum principle covering the locally Lipschitz case.