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R. Acad. Sci. Paris Ser. I Math., 327 (1998), pp. 735-741]. The corresponding Hamilton-Jacobi-Bellman (HJB) equation, which turns out to be a non-adapted stochastic partial differential equation, is analyzed. Making use of the viscosity solution framework, we show that the value function of the optimal control problem is the unique solution of the HJB equation. When the optimal drift is defined, we provide its characterization. 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