On the path structure of a semimartingale arising from monotone probability theory

TL;DR

This paper studies a specific semimartingale arising from monotone probability theory, analyzing its path structure, level sets, and local times, revealing unique probabilistic properties and behaviors.

Contribution

It introduces and analyzes the path structure of a monotone-independent semimartingale, highlighting its level set properties and local time characteristics.

Findings

The level set where Y_t=1 is non-empty, compact, perfect, and has zero Lebesgue measure.

Local times are trivial except at level 1.

Jumps of Y are not locally summable.

Abstract

Let $X$ be the unique normal martingale such that $X_0=0$ and \[\mathrm{d}[X]_t=(1-t-X_{t-}) \mathrm{d}X_t+\mathrm{d}t\] and let $Y_t:=X_t+t$ for all $t\geq 0$; the semimartingale $Y$ arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of $Y$ are examined and various probabilistic properties are derived; in particular, the level set $\{t\geq 0\dvt Y_t=1\}$ is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of $Y$ are found to be trivial except for that at level 1; consequently, the jumps of $Y$ are not locally summable.