  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.
