Some properties of Solutions of stochastic differential equations driven by semi-martingales

In this paper, we study stochastic differential equations driven by semi-martingales, with time dependent and non Lipschitz coefficients. The properties established are based on an improvement of Krylov's inequality for semi-martingales. Under a condition of absolute continuity of the quadratic Variation process associated to the martingale part, we prove that Krylov's inequality remains valid for functions which depends explicitely on the time variable. This last inequality is then applied to derive various properties such äs pathwise uniqueness, non confluent property and continuity with respect to initial data for stochastic differential equations driven by continuous semi-martingales with Sobolev space valued coefficients. Exponential martingales will be the main tool in the proois.