From Markov Processes to Semimartingales

TL;DR

This paper reviews the historical development and connections between Markov processes and semimartingales, highlighting how key concepts evolved and transferred between these frameworks in stochastic process theory.

Contribution

It provides a comprehensive overview of the historical and conceptual links between Markov processes and semimartingales, emphasizing the transfer of ideas and concepts.

Findings

Semimartingales are the natural class for many concepts originally developed for Markov processes.

Key concepts like stochastic differential equations and killing were developed in Markov process theory before being incorporated into semimartingales.

The paper discusses symbols, characteristics, and Blumenthal-Getoor indices in the context of these processes.

Abstract

In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the `natural' class of processes for many concepts first developed in the Markovian framework. As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently. We describe, when these and other important concepts have been invented in the theory of Markov processes and how they were transferred to semimartingales. Further topics include the symbol, characteristics and generalizations of Blumenthal-Getoor indices. Some additional comments on relations between Markov processes and semimartingales round out the paper.