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On Watanabe's characterisation and change of intensity à la Girsanov for Cox processes

Dirk Becherer, Thomas Bernhardt, Pavel Gapeev

TL;DR

The paper addresses the fundamental question of when conditional Poisson processes, Cox processes, and stochastic intensities are equivalent definitions for simple point processes on the real line. It provides an elementary proof of the Watanabe-type equivalence, derives explicit conditional jump-time densities, and shows the compensated process \widetilde N is a local martingale. It then develops a Girsanov-type change of measure via a martingale exponential to transform the intensity from X to a target YX, with conditions ensuring the exponential is a true martingale. The results underpin the reference probability approach to filtering, illustrating how to implement intensity changes in practice and relating to existing martingale criteria (e.g., Kallsen–Karbe).

Abstract

We discuss the equivalence of definitions for conditional Poisson processes, Cox processes, and stochastic intensities of point processes on the real line. We show that Watanabe's characterisation of conditional Poisson processes in terms of local martingales is necessary and sufficient. Additionally, we consider conditions enabling the measure change method a la Girsanov to alter the intensity of Cox processes to a desired new target intensity, e.g. for the probability reference approach in filtering. Such a measure change exists if a corresponding stochastic exponential is a proper martingale. We show that this holds if the new locally integrable target intensity is the product of the original intensity and another non-negative process.

On Watanabe's characterisation and change of intensity à la Girsanov for Cox processes

TL;DR

The paper addresses the fundamental question of when conditional Poisson processes, Cox processes, and stochastic intensities are equivalent definitions for simple point processes on the real line. It provides an elementary proof of the Watanabe-type equivalence, derives explicit conditional jump-time densities, and shows the compensated process \widetilde N is a local martingale. It then develops a Girsanov-type change of measure via a martingale exponential to transform the intensity from X to a target YX, with conditions ensuring the exponential is a true martingale. The results underpin the reference probability approach to filtering, illustrating how to implement intensity changes in practice and relating to existing martingale criteria (e.g., Kallsen–Karbe).

Abstract

We discuss the equivalence of definitions for conditional Poisson processes, Cox processes, and stochastic intensities of point processes on the real line. We show that Watanabe's characterisation of conditional Poisson processes in terms of local martingales is necessary and sufficient. Additionally, we consider conditions enabling the measure change method a la Girsanov to alter the intensity of Cox processes to a desired new target intensity, e.g. for the probability reference approach in filtering. Such a measure change exists if a corresponding stochastic exponential is a proper martingale. We show that this holds if the new locally integrable target intensity is the product of the original intensity and another non-negative process.
Paper Structure (2 sections, 3 theorems, 22 equations)

This paper contains 2 sections, 3 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Statements and Proofs

Key Result

Lemma 2.1

Let $N$ be a conditional Poisson process with $\mathcal{F}_0$-measurable intensity $X$, let $Y$ be a $\,\mathcal{B}([0,\infty])\otimes\mathcal{F}_r$-measurable non-negative process for given $\,r\geq0$, and $\,n\in\mathbb{N}_0$. Then with conditional density $\,\Psi^{n,r}\colon[0,\infty]\times\Omega\to[0,\infty)\,$ given by using the conventions $\,0^0=1\,$ and $\,\mathrm{e}^{-\infty}\infty=0$.

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Lemma 2.1: Conditional Density
  • Theorem 2.2: Characterisation à la Watanabe1964 and Grigelionis1975
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Changing intensity by change of measure
  • Remark 2.6
  • Example 2.7: Filtering compound Poisson processes from observations about the jump times