Optimal Novikov-type criteria for local martingales with jumps. (English) Zbl 1312.60051

Sokol, Alexander

Electron. Commun. Probab. 18, Paper No. 39, 8 p. (2013).

Summary: We consider cadlag 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 an 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.

MSC:

60G44	Martingales with continuous parameter
60G40	Stopping times; optimal stopping problems; gambling theory

Keywords:

martingales; uniform integrability; Novikov-type criteria; Poisson process