On controlled diffusion processes with unbounded coefficients

The paper is devoted to the general theory of controlled diffusion processes in a domain of a ¿/-dimensional space in the absence of constraints on the growth of the coefficients at infinity. It turned out that the most suitable object of study is the payoff function in the optimal stopping problem for the controlled process. A theory analogous to the theory of controlled processes in the whole space with growth constraints on the coefficients is developed under natural assumptions.