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

Belton, Alexander C. R.

Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 2, pp. 258-279.

Let $X$ be the unique normal martingale such that $X_0=0$ and

$$\mathrm{d}{\left[X\right]}_t=(1-t-X_{t-})\mathrm{d}{X}_t+\mathrm{d}t$$

and let $Y_t:=X_t+t$ for all $t\ge 0$; 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 $\left\{t\ge 0:Y_t=1\right\}$ 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.

Soit $X$ l’unique martingale normale telle que $X_0=0$ et

$$\mathrm{d}{\left[X\right]}_t=(1-t-X_{t-})\mathrm{d}{X}_t+\mathrm{d}t$$

et soit $Y_t:=X_t+t$ pour tout $t\ge 0$; la semimartingale $Y$ se manifeste dans la théorie des probabilités quantiques, où c’est analogue du processus de Poisson pour l’indépendance monotone. Les trajectoires de $Y$ sont examinées et diverses propriétés probabilistes sont déduites; en particulier, l’ensemble de niveau $\left\{t\ge 0:Y_t=1\right\}$ est montré être non vide, compact, parfait et de mesure de Lebesgue nulle. Les temps locaux de $Y$ sont trouvés être triviaux sauf celui au niveau 1; par conséquent les sauts de $Y$ ne sont pas localements sommables.

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