On the Semimartingale Nature of Feller Processes with Killing

TL;DR

This paper demonstrates that Feller processes with killing are semimartingales under mild conditions, providing explicit characteristics and a formula to compute their symbols, thus bridging stochastic process theory and probabilistic analysis.

Contribution

It generalizes semimartingale theory to processes with killing and explicitly relates their characteristics to the process's symbol.

Findings

Feller processes with killing are semimartingales under mild assumptions

Explicit calculation of semimartingale characteristics for these processes

Probabilistic formula for directly computing the process's symbol

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 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 directly.