On the semimartingale nature of Feller processes with killing

Abstract

Let U be an open set in R d . We show that under a mild assumption on the richness of the generator a Feller process in (U, B(U )) with (predictable) killing is a semimartingale. To this end we generalize the notion of semimartingales in a natural way to those 'with killing'. Furthermore we calculate the semimartingale characteristics of the Feller process explicitly and analyze their connections to the symbol. Finally we derive a probabilistic formula to calculate the symbol of the process.MSC 2010: 60J75 (primary); 60J25, 60H05, 47G30, 60G51 (secondary).

The fourth characteristic of a semimartingale

“…The following results encompases [35] Theorem 4.3. It is easily deduced combining Theorem 2.13, Corollary 2.14 respectively its proof and Theorem 2.18: Theorem 3.3.…”

“…Let us mention that this very simple class of Lévy processes with killing is not included in the classical semimartingale setting. In [35] we have introduced a class of semimartingales admitting a predictable killing, but even in this framework Lévy processes with killing are not included. This is somehow not satisfactory since they perfectly fit into the framework of sub-Markovian kernels and hence Markov processes which are in turn closely linked to semimartingales (cf.…”

“…Since the three characteristics have become canonical, let us give a second motivation, why it is useful to introduce a fourth characteristic. There is an intimate relationship between Markov processes and semimartingales which has been studied in [9], [35] and [8]. Let us elaborate on this relationship in the case of Feller processes with sufficiently rich domain.…”

“…Usually it is directly assumed that a = 0 (cf. [35], [30], [4], [5]). Hence, nonpredictable killing is excluded.…”

Space‐time fractional Dirichlet problems

“…Remark Note that the equality for suitable functions f is proved in [, Corollary 3.8] under the assumption , which do not apply in our case, see Remark .…”