Infinite horizon and ergodic optimal quadratic control for an affine equation with stochastic coefficients

We study stochastic optimal control problems with a quadratic cost and linear state equation that involves stochastic coefficients and control dependent noise and, moreover, is perturbed by an affine term. Both the infinite horizon case and the ergodic case are treated. To this purpose we introduce a backward stochastic Riccati equation and a dual backward stochastic equation, both considered in the whole time line. Besides some stabilizability conditions we prove the existence of a solution for the two previous equations defined as limit of suitable finite horizon approximating problems. This allows us to perform the synthesis of the optimal control.