Pathwise uniqueness of stochastic differential equations driven by Brownian motions and finite variation Levy processes

We study the pathwise uniqueness of the solutions to one-dimensional stochastic differential equations driven by Brownian motions and Levy processes with finite variation paths. The driving Levy processes are not necessarily one-sided jump processes. In this paper, we obtain some non-Lipschitz conditions on the coefficients, under which the pathwise uniqueness of the solution to the equations is established. Some of our results can be applied to the equation with discontinuous coefficients.