The Fourth Characteristic of a Semimartingale

TL;DR

This paper generalizes the concept of semimartingales to include processes with paths leaving the state space, introducing a fourth characteristic that extends the classical Lévy quadruple, and analyzing their generators via probabilistic symbols.

Contribution

It introduces a new fourth semimartingale characteristic for processes with exit states, broadening the classical framework and linking it to generators through probabilistic symbols.

Findings

Defined a new fourth characteristic for semimartingales with exit states.

Connected the new characteristic to generators of Markov processes with killing.

Extended the classical Lévy quadruple framework.

Abstract

We extend the class of semimartingales in a natural way. This allows us to incorporate processes having paths that leave the state space R^d. In particular Markov processes related to sub-Markovian kernels, but also non-Markovian processes with path-dependent behavior. By carefully distinguishing between two killing states, we are able to introduce a fourth semimartingale characteristic which generalizes the fourth part of the L\'evy quadruple. Using the probabilistic symbol, we analyze the close relationship between the generators of certain Markov processes with killing and their (now four) semimartingale characteristics.