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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 thecorresponding exponential local martingale to be a uniformly integrablemartingale. We obtain criteria using both the quadratic variation and thepredictable quadratic variation. We prove optimality of the coefficients in thecriteria. As a corollary, we obtain a verbatim extension of the classicalNovikov criterion for continuous local martingales to the case of localmartingales with nonnegative jumps.

Optimal Novikov-type criteria for local martingales with jumps