Strong completeness for a class of stochastic differential equations with irregular coefficients

We prove the strong completeness for a class of non-degenerate SDEs, whose coefficients are not necessarily uniformly elliptic nor locally Lipschitz continuous nor bounded. Moreover, for each p > 0 there is a positive number T(p) such that for all t < T(p), the solution flow F-t(.) belongs to the Sobolev space W-loc(1,p). The main tool for this is the approximation of the associated derivative flow equations. As an application a differential formula is also obtained.