Perturbed Linear-Quadratic Control Problems and Their Probabilistic Representations

We consider some certain nonlinear perturbations of the stochastic linear-quadratic optimization problems and study the connections between their solutions and the corresponding Markovian backward stochastic diferential equations (BSDEs). Using the methods of stochastic control, nonlinear partial differential equations (PDEs) and BSDEs, we identify conditions for the solvability of the problem and obtain some regularity properties of the solutions.