In modeling dynamical systems, the stochastic framework is suitable for taking into account either randomly varying system parameters or stochastic exogenous inputs. This paper considers nonlinear dynamical systems, consisting of a linear nominal part plus model uncertainties, nonlinearities, and both additive and multiplicative random noise, modeled as a Wiener process. In particular, the problem of finding suitable measurement feedback control laws is studied. These laws are such that the resulting closed-loop system is stable in some probabilistic sense and a given cost functional is minimized. A Lyapunov-based separation result is given, which splits the control design into a state feedback problem and a filtering problem. The constructive algorithms for solving the state feedback and filtering problems with an arbitrarily large region of attraction for a wide class of nonlinear systems, which include feedback linearizable systems, are presented.