Abstract: This paper investigates the linear minimum mean square error state estimation for discrete-time systems with Markov jump delays. In order to solve the optimal estimation problem, the single Markov delayed measurement is rewritten as an equivalent measurement with multiple constant delays, then a delay-free Markov jump linear system is obtained via state augmentation. The estimator is derived on the basis of the geometric arguments in the Hilbert space, and a recursive equation of the filter is obtained by solving the Riccati equations. It is shown that the proposed state estimator is exponentially stable under standard assumptions.