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This paper deals with prediction of controlled autoregressive processes with additive white Gaussian noise and random coefficients adapted to an observation process. Our aim is twofold. We begin by extending to the standard Kalman predictor a result of Chen et al. (1989) on the optimality of the "standard Kalman filter" when applied to linear stochastic processes with almost surely finite random coefficients. ; We then show on an example how some particular nonlinear autoregressive processes can be embedded in these linear processes with random coefficients. Such nonlinear processes can then benefit from this optimal prediction, which is not provided by the usual "extended Kalman predictor".