Optimal Novikov-type criteria for local martingales with jumps

We consider càdlàg local martingales M with initial value zero and jumps larger than a for some a larger than or equal to −1, and prove Novikov-type criteria for the exponential local martingale to be a uniformly integrable martingale. We obtain criteria using both the quadratic variation and the predictable quadratic variation. We prove optimality of the coefficients in the criteria. As a corollary, we obtain a verbatim extension of the classical Novikov criterion for continuous local martingales to the case of local martingales with initial value zero and nonnegative jumps.