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

Belton, Alexander C. R. (2008) On the path structure of a semimartingale arising from monotone probability theory. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 44 (2). pp. 258-279. ISSN 0246-0203

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Abstract

Let X be the unique normal martingale such that X_0 = 0 and d[X]_t = (1 - t - X_{t-}) dX_t + dt and let Y_t := X_t + t for all t >= 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 >= 0 : 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.

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