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We study a class of systems whose parameters are driven by a Markov chain in reverse time. A recursive characterization for the second moment matrix and the formulas for optimal control are given. Our results are determining the answer for the question: Is it possible to extend the classical duality between filtering and control of linear systems (whose matrices are transposed in the dual problem) by simply adding the jump variable of a Markov jump linear system? The answer is positive provided the jump process is reversed in time.

On the Linear Quadratic Problem for Systems With Time-Reversed Markov Jump Parameters and the Duality With Filtering of Markov Jump Linear Systems