On the stochastic integral equations with non-Lipschitz coefficients

Consider the stochastic integral equation (S.I.E.) X(t) = H(t) + integral(0+)(t) f(X(s(-)))dZ(s), t is an element of R+ (0.1) where f satisfies some non-Lipschitz condition and H,Z are F-t-semimartingales, continuous or discontinuous, on some probability space (Omega, F, {F-t}(tis an element ofR+), P). We prove that if f satisfies Condition H-1 or H-2 (defined in Sec. 0), then both the existence and the uniqueness of the solutions of (0.1) hold.