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On minimal predictable intensity of point processes

Haoming Wang

TL;DR

The paper addresses when a simple point process admits a minimal predictable intensity, proving this occurs precisely when the process is a standard Poisson process under an absolutely continuous change of measure. It introduces a function-analytic approach using the Hahn-Banach theorem to construct a unique $oldsymbol{G}$-predictable intensity ${}^p\lambda$ that makes $N - A({}^p\lambda)$ a martingale under a transformed measure $\boldsymbol{Q}$, without requiring null completeness. It also delivers a canonical Lebesgue decomposition for Cox processes and clarifies the relationship between Hawkes processes and Poisson processes, notably showing Hawkes reduce to Poisson only when self-excitation vanishes. Together, these results provide a rigorous minimal-filtration framework for representing point processes under measure changes with practical implications for Cox/Hawkes modeling and time-change analyses.

Abstract

An adapted, right-continuous, non-decreasing, integer-valued process with unit jumps and starting at zero has a minimal predictable intensity if and only if it is a standard Poisson process under an absolutely continuous transformation of measures.

On minimal predictable intensity of point processes

TL;DR

The paper addresses when a simple point process admits a minimal predictable intensity, proving this occurs precisely when the process is a standard Poisson process under an absolutely continuous change of measure. It introduces a function-analytic approach using the Hahn-Banach theorem to construct a unique -predictable intensity that makes a martingale under a transformed measure , without requiring null completeness. It also delivers a canonical Lebesgue decomposition for Cox processes and clarifies the relationship between Hawkes processes and Poisson processes, notably showing Hawkes reduce to Poisson only when self-excitation vanishes. Together, these results provide a rigorous minimal-filtration framework for representing point processes under measure changes with practical implications for Cox/Hawkes modeling and time-change analyses.

Abstract

An adapted, right-continuous, non-decreasing, integer-valued process with unit jumps and starting at zero has a minimal predictable intensity if and only if it is a standard Poisson process under an absolutely continuous transformation of measures.
Paper Structure (5 sections, 17 theorems, 36 equations, 2 figures)

This paper contains 5 sections, 17 theorems, 36 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and Main Results
  3. Proof of Main Theorem \ref{['thm: main theorem 1']}
  4. Proof of the Converse of Main Theorem \ref{['thm: main theorem 1']}
  5. Applications to the Cox and Hawkes Processes

Key Result

Theorem 1

Suppose $\underline{\underline{N}} = \{N(\omega,t),\boldsymbol{F}, \boldsymbol{P}\}$ is a Poisson process of rate 1 and $\lambda(\omega,t)$ satisfies (A0)-(A4). Take $t = \infty$ in eq: girsanov transformation. We assume that $\boldsymbol{Q}(\boldsymbol{\varOmega}) = 1.$ For any $\boldsymbol{G}$ suc

Figures (2)

  • Figure 1: The intensity $\lambda_{t}$ over time $t$ in a path of Hawkes process under two scenarios (a) $\mu(t) = 0.3 + 0.2 e^{-0.1t}$ and (b) $\mu(t) = 0.1 + 0.2 e^{-0.1t}$ and $\phi(t) = 0.2$.
  • Figure :

Theorems & Definitions (32)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Lemma 1
  • proof : Proof of Lemma \ref{['thm: watanabe-bremaud']}
  • Lemma 2
  • proof : Proof of Lemma \ref{['lem: exp martingale']}
  • Lemma 3
  • Lemma 4
  • proof : Proof of Lemma \ref{['lem: compensated martingale']}
  • ...and 22 more