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We present an adaptive output feedback controller for a class of uncertain stochastic nonlinear systems. The plant dynamics is represented as a nominal linear system plus nonlinearities. In turn, these nonlinearities are decomposed into a part, obtained as the best approximation given by neural networks, plus a remaining part which is treated as uncertainties, modeling approximation errors, and neglected dynamics. The weights of the neural network are tuned adaptively by a Lyapunov design. The proposed controller is obtained through robust optimal design and combines together parameter projection, control saturation, and high-gain observers. High performances are obtained in terms of large errors tolerance as shown through simulations.