Abstract: | In this paper, we consider the following problem: Here the coefficients aij and bi are smooth, periodic with respect to the second variable, and the matrix (aij)ij is uniformly elliptic. The Hamiltonian H is locally Lipschitz continuous with respect to u? and Du?, and has quadratic growth with respect to Du?. The Hamilton-Jacobi-Beliman equations of some stochastic control problems are of this type. Our aim is to pass to the limit in (0?) as ? tends to zero. We assume the coefficients bi to be centered with respect to the invariant measure of the problem (see the main assumption (3.13)). Then we derive L∞, H and W , p0 > 2, estimates for the solutions of (0?). We also prove the following corrector's result: This allows us to pass to the limit in (0?) and to obtain This problem is of the same type as the initial one. When (0?) is the Hamilton-Jacobi-Bellman equation of a stochastic control problem, then (00) is also a Hamilton-Jacobi-Bellman equation but one corresponding to a modified set of controls. |