Structure of infinitely divisible semimartingales

TL;DR

This paper provides a comprehensive characterization and decomposition of infinitely divisible semimartingales, extending classical Gaussian results and analyzing their jump structures and stationary increments.

Contribution

It introduces a new approach combining series decompositions and jump analysis to characterize infinitely divisible semimartingales explicitly.

Findings

Explicit decomposition of infinitely divisible semimartingales

Characterization of semimartingale property for stationary increment processes

Extension of classical Gaussian semimartingale theorems

Abstract

This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such semimartingales is obtained. A new approach, combining series decompositions of infinitely divisible processes with detailed analysis of their jumps, is presented. As an ilustration of the main result, the semimartingale property is explicitely determined for a large class of stationary increment processes and several examples of processes of interest are considered. These results extend Stricker's theorem characterizing Gaussian semimartingales and Knight's theorem describing Gaussian moving average semimartingales, in particular.