Convergence of local supermartingales and Novikov-Kazamaki type conditions for processes with jumps

TL;DR

This paper characterizes the convergence of local supermartingales with jumps using predictable characteristics and quadratic variation, and applies these results to establish Novikov-Kazamaki type conditions for martingale properties.

Contribution

It introduces new criteria for convergence of local supermartingales with jumps and provides a novel proof for Novikov-Kazamaki conditions in this context.

Findings

Characterization of local supermartingale convergence

Conditions in terms of predictable characteristics and quadratic variation

Proof of Novikov-Kazamaki conditions for jump processes

Abstract

We characterize the event of convergence of a local supermartingale. Conditions are given in terms of its predictable characteristics and quadratic variation. The notion of extended local integrability plays a key role. We then apply these characterizations to provide a novel proof for the sufficiency and necessity of Novikov-Kazamaki type conditions for the martingale property of nonnegative local martingales with jumps.