Existence of Optimal Controls for Variance Control

The question of the existence of optimal controls is fundamental to many further developments in control theory, and there is a huge literature on the subject. For stochastic control, variance control problems have received much attention recently owing to their importance in nancial mathematics. The existence question is answered in much the same way as it is when only the drift is controlled. In order to establish the intuition, classical results are surveyed and the necessary extensions for the variance control problem are outlined.