Piecewise constant local martingales with bounded numbers of jumps

TL;DR

This paper characterizes when a piecewise constant local martingale with limited jumps is a uniformly integrable martingale, showing it depends on the integrability of its negative part at infinity.

Contribution

It provides a necessary and sufficient condition for such martingales to be uniformly integrable, linking it to the integrability of the negative terminal value.

Findings

Uniform integrability depends on the integrability of the negative part at infinity.

Characterization applies to piecewise constant local martingales with bounded jumps.

Provides a clear criterion for martingale property in this class.

Abstract

A piecewise constant local martingale $M$ with boundedly many jumps is a uniformly integrable martingale if and only if $M_\infty^-$ is integrable.