  We consider local martingales $M$ with jumps larger than $a$ for some $a$
larger than or equal to -1, and prove Novikov-type criteria for the
corresponding 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 nonnegative jumps.
