Markov property of monotone L\'evy processes

Abstract

Monotone L\'evy processes with additive increments are defined and studied. It is shown that these processes have a natural Markov structure and their Markov transition semigroups are characterized using the monotone L\'evy-Khintchine formula. Monotone L\'evy processes turn out to be related to classical L\'evy processes via Attal's ``remarkable transformation.'' A monotone analogue of the family of exponential martingales associated to a classical L\'evy process is also defined.