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On the extension and kernels of signed bimeasures and their role in stochastic integration

Riccardo Passeggeri

TL;DR

The paper develops a general extension theory for signed bimeasures on $\delta$-rings, providing a necessary-and-sufficient condition that yields a unique finite signed measure with relative kernels and a density $q$, extending Horowitz and Rajput–Rosinski constructions to the signed setting. Building on this foundation, it then constructs the most general stochastic integration theory for quasi-infinitely divisible (QID) random measures, unifying and extending the Rajput–Rosinski (1989) and Passeggeri (2020) frameworks under a single broad assumption. The resulting theory gives a Lévy–Khintchine type representation for $\Lambda(A)$, explicit integrability criteria, and a continuous integral mapping in Musielak–Orlicz spaces, delivering the complete integration theory for real-valued random measures. This work broadens the modeling toolkit for stochastic processes, with potential applications in Bayesian nonparametric analysis and stochastic process representations where signed measures and QID distributions arise.

Abstract

In this work we provide a necessary and sufficient condition for the extension of signed bimeasures on $δ$-rings and for the existence of relative kernels. This result generalises the construction method of regular conditional probabilities to the more general setting of extended signed measures. Building on this result, we obtain the most general theory of stochastic integrals based on random measures, thus extending and generalising the whole integration theory developed in the celebrated Rajput and Rosinski's paper (\textit{Probab.~Theory Relat.~Fields}, \textbf{82} (1989) 451-487) and the recent results by Passeggeri (\textit{Stoch.~Process.~Their Appl.}, \textbf{130}, (3), (2020), 1735-1791).

On the extension and kernels of signed bimeasures and their role in stochastic integration

TL;DR

The paper develops a general extension theory for signed bimeasures on -rings, providing a necessary-and-sufficient condition that yields a unique finite signed measure with relative kernels and a density , extending Horowitz and Rajput–Rosinski constructions to the signed setting. Building on this foundation, it then constructs the most general stochastic integration theory for quasi-infinitely divisible (QID) random measures, unifying and extending the Rajput–Rosinski (1989) and Passeggeri (2020) frameworks under a single broad assumption. The resulting theory gives a Lévy–Khintchine type representation for , explicit integrability criteria, and a continuous integral mapping in Musielak–Orlicz spaces, delivering the complete integration theory for real-valued random measures. This work broadens the modeling toolkit for stochastic processes, with potential applications in Bayesian nonparametric analysis and stochastic process representations where signed measures and QID distributions arise.

Abstract

In this work we provide a necessary and sufficient condition for the extension of signed bimeasures on -rings and for the existence of relative kernels. This result generalises the construction method of regular conditional probabilities to the more general setting of extended signed measures. Building on this result, we obtain the most general theory of stochastic integrals based on random measures, thus extending and generalising the whole integration theory developed in the celebrated Rajput and Rosinski's paper (\textit{Probab.~Theory Relat.~Fields}, \textbf{82} (1989) 451-487) and the recent results by Passeggeri (\textit{Stoch.~Process.~Their Appl.}, \textbf{130}, (3), (2020), 1735-1791).

Paper Structure

This paper contains 4 sections, 15 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Extension of signed bimeasures and existence of relative kernels
  4. Stochastic integration based on random measures

Key Result

Theorem 2.4

Let $\mu$ be a measure on a ring $\mathcal{R}$ of subsets of a space $X$. Assume that $\mu$ is $\sigma$-finite (i.e. that there exists $S_{1},S_{2},...\in\mathcal{R}$ such that $X=\cup_{n=1}^{\infty}S_{n}$ and that $\mu(S_{n})<\infty$ for every $n\in\mathbb{N}$) then there exists a unique $\sigma$-f

Theorems & Definitions (42)

  • Definition 2.1: extended signed measure
  • Definition 2.2: Signed bimeasure
  • Definition 2.3: Signed measure on a ring
  • Theorem 2.4: Carathéodory's extension theorem, see Theorem 1.41 in Klenke
  • Definition 2.5: random measure
  • Theorem 2.6: Theorem 2.1 in PrekopaI
  • Definition 2.7
  • Definition 2.8: quasi-Lévy type measure, quasi-Lévy measure, QID distribution, from LPS
  • Definition 2.9: QID random measure
  • Remark 2.10
  • ...and 32 more