Citation Link: https://doi.org/10.1214/23-ps19
From Markov processes to semimartingales
Publication Type
Article
Subjects
Mathematics
Markov process
Markov chain
Markov property
Markov kernel
Variable-order Markov model
Class (philosophy)
Applied mathematics
Markov model
Mathematical economics
Computer science
Artificial intelligence
Statistics
Date Issued
2023
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 where 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.
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