On criteria for the existence of the strong solution of the stochastic equation

Criteria for the existence of the strong solution and for the strong uniqueness of a solution of the Ito's stochastic differential equation
$$ dx_t=\sigma(t,x_t)\,dw_t+b(t,x_t)\,dt,\qquad x_0=x\in E_d, $$
are formulated in terms of the linear parabolic equations theory and proved.