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In this paper, we study the optimal control problem of a controlledtime-symmetric forward-backward doubly stochastic differential equation withinitial-terminal sate constraints. Applying the terminal perturbation methodand Ekeland's variation principle, a necessary condition of the stochasticoptimal control, i.e., stochastic maximum principle is derived. Applications tobackward doubly stochastic linear-quadratic control models are investigated.

A maximum principle for controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal sate constraints