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This paper is devoted to establish continuous dependence estimates for theergodic problem for Bellman operators (namely, estimates of (v_1-v_2) where v_1and v_2 solve two equations with different coefficients). We shall obtain anestimate of ||v_1-v_2||_\infty with an explicit dependence on theL^\infty-distance between the coefficients and an explicit characterization ofthe constants and also, under some regularity conditions, an estimate of||v_1-v_2||_{C^2(\R^n)}.  Afterwards, the former result will be crucial in the estimate of the rate ofconvergence for the homogenization of Bellman equations. In some regular cases,we shall obtain the same rate of convergence established in the monographs[11,26] for regular linear problems.

Continuous dependence estimates for the ergodic problem of Bellman  equation with an application to the rate of convergence for the  homogenization problem