Maximum principle for BSDEs with locally Lipschitz and logarithmic growth

This paper tackles a stochastic control problem involving a backward stochastic differential equation (BSDE) with a local Lipschitz coefficient and logarithmic growth. We derive the necessary and sufficient conditions for optimality that hold for all optimal controls, even without convexity assumptions on the control domain. These conditions involve a local Lipschitz stochastic differential equation and a minimized Hamiltonian. We begin by demonstrating the existence and uniqueness of the solution to the associated adjoint equation under suitable conditions. Next, we introduce a series of control problems with global Lipschitz coefficients using an approximation approach. This framework allows us to derive a stochastic maximum principle, facilitating the analysis of near-optimal controls within these approximated systems. Finally, we seamlessly transition back to the initial control problem through a well-defined limit process.