A kind of stochastic optimization problem solved by the BSDE method (Q2774102)

A kind of stochastic optimization problem related to a backward stochastic equation (BSDE) NEWLINE\[NEWLINE-dx_t= f(x_t,z_t,t) dt-z_tdW_t,\;x_T= \xi\;(t \in[0,T])NEWLINE\]NEWLINE is considered here, where the optimal objective \(\xi^*\) means that the functional \(J(\xi^*)\) reaches its maximum with respect to any other permissible objective [for the problem and the related notations, cf. \textit{E. Pardoux} and \textit{S. Peng}, Syst. Control Lett. 14, 55-61 (1990; Zbl 0692.93064)]. First, the authors introduce a variational method for the end condition of the BSDE. Then, without the assumption of convex properties about the coefficient of the equation, as usually used by others, a necessary condition (or Maximum Principle) for the optimal objective is obtained using the Ekeland variational principle.