Optimal Novikov-type criteria for local martingales with jumps

TL;DR

This paper establishes optimal Novikov-type criteria for exponential local martingales with jumps, extending classical results to cases with jumps larger than a specified threshold, using quadratic variation measures.

Contribution

It introduces the first optimal Novikov-type criteria for local martingales with jumps larger than a threshold, extending classical continuous martingale results.

Findings

Criteria using quadratic variation and predictable quadratic variation

Optimality of the coefficients in the criteria

Extension of Novikov criterion to nonnegative jump martingales

Abstract

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.