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

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.