The paper deals with Markov jump linear system driven by wide-sense stationary noise, stabilizable in the mean square sense by linear feedback controls, which may or my not depend on the observation of the underlying Markov jump state. The main result is an evaluation that connects the finite and the long run average costs in terms of a two-sided bound for the former cost. The derived evaluation allows us to conclude straightforwardly on the existence of the long run average cost, and hence, on the existence of the optimal control solution. For given initial condition and control, the evaluation also can be faced as an error bound for the approximation of the long run average cost by associate finite-horizon costs, thus setting an initial landmark on approximation techniques. © 2006 IEEE. 55435548Costa, O.L.V., Fragoso, M.D., Marques, R.P., (2005) Discrete-Time Markovian Jump Linear Systems, , New York: Springer-VerlagCosta, O.L.V., Fragoso, M.D., Stability results for discrete-time linear systems with Markovian jumping parameters (1993) Journal of Mathematical Analysis and Aplications, 179, pp. 154-178Costa, O.L.V., Fragoso, M.D., Discrete-time LQ-optimal control problems for finite Markov jump parameters systems (1995) IEEE Transactions on Automatic Control, 40, pp. 2076-2088Ji, Y., Chizeck, H.J., Controllability, stabilizability and continuous-time Markovian jump linear quadratic control (1990) IEEE Transactions on Automatic Control, 35 (7), pp. 777-788Ji, Y., Chizeck, H.J., Jump linear quadratic Gaussian control: Steady-state solution and testable conditions (1990) Control-Theory and Advanced Technology, 6 (3), pp. 289-319. , SeptemberCosta, E.F., do Val, J.B.R., Fragoso, M.D., A new approach to detectability of discrete-time Markov jump linear systems (2005) SIAM Journal on Control and Optimization, 43 (6), pp. 2132-2156Hernandez-Lerma, O., Hennet, J.C., Lasserre, J.B., Average cost Markov decision processes: Optimality conditions (1991) Journal of Mathematical Analysis and Applications, 158, pp. 396-406Hernandez-Lerma, O., Lasserre, J.-B., (1996) Discrete-Time Markov Control Processes: Basic Optimality Criteria, , SpringerRoss, S., (1970) Applied Probability Models with Optimization Applications, , Holden-DayVargas, A.N., do Val, J.B.R., Costa, E.F., Receding horizon control of Markov jump linear systems subject to noise and unobservable state chain (2004) 43th IEEE Conference on Decision and Control, pp. 4381-4386. , Paradise Island, The Bahamas