Stochastic equations and krylov's estimates for semimartingales

In 1969 N. V. Krylov obtained the following estimate where X is an Ito process in is the exit time of X from a bounded region is Ld-norm of a measurable nonnegative function , and N is a constant. We generalize estimates of this type to semimartingales and give applications to the theory of stochastic equations with respect to semimartingales. The questions of the existence, uniqueness, convergence and comparison of solutions of these equations are also studied.