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A Global Maximum Principle for the Stochastic Optimal Control Problem with Delay

In this paper, an open problem is solved, for the stochastic optimal control problem with delay where the control domain is nonconvex and the diffusion term contains both control and its delayed term. Inspired by previous results by Øksendal and Sulem [{\it A maximum principle for optimal control of stochastic systems with delay, with applications to finance. In J. M. Menaldi, E. Rofman, A. Sulem (Eds.), Optimal control and partial differential equations, ISO Press, Amsterdam, 64-79, 2000}] and Chen and Wu [{\it Maximum principle for the stochastic optimal control problem with delay and application, Automatica, 46, 1074-1080, 2010}], Peng's general stochastic maximum principle [{\it A general stochastic maximum principle for optimal control problems, SIAM J. Control Optim., 28, 966-979, 1990}] is generalized to the time delayed case, which is called the global maximum principle. A new backward random differential equation is introduced to deal with the cross terms, when applying the duality technique. Comparing with the classical result, the maximum condition contains an indicator function, in fact it is the characteristic of the stochastic optimal control problem with delay. The multi-dimensional case and a solvable linear-quadratic example are also discussed.