[Paper Review] Properties of Doubly Stochastic Poisson Process with affine intensity

Alan Genaro, Adilson Simonis|arXiv (Cornell University)|Sep 13, 2011

Stochastic processes and financial applications15 references3 citations

TL;DR

This paper derives analytical expressions for the probability distribution functions of a Doubly Stochastic Poisson Process (DSPP) with affine intensity, enabling exact likelihood-based inference. It applies the framework to a one-dimensional Feller process intensity, estimating parameters via Kalman filtering on high-frequency transaction data, providing a tractable model for financial point processes.

ABSTRACT

This paper discusses properties of a Doubly Stochastic Poisson Process (DSPP) where the intensity process belongs to a class of affine diffusions. For any intensity process from this class we derive an analytical expression for probability distribution functions of the corresponding DSPP. A specification of our results is provided in a particular case where the intensity is given by one-dimensional Feller process and its parameters are estimated by Kalman filtering for high frequency transaction data.

Motivation & Objective

• To develop a tractable analytical framework for the probability distribution of a Doubly Stochastic Poisson Process (DSPP) when the intensity follows an affine diffusion process.
• To extend existing point process models by incorporating affine intensity dynamics, enabling exact likelihood computation.
• To apply the theoretical framework to high-frequency financial transaction data using parameter estimation via Kalman filtering.
• To validate the model's empirical relevance by estimating parameters of a Feller process intensity in a real-world financial context.

Proposed method

• Model the intensity process as an affine diffusion, ensuring analytical tractability of the moment-generating function.
• Derive the probability distribution function of the DSPP using the Laplace transform of the integrated intensity.
• Utilize the affine structure to express the moment-generating function in closed form, enabling exact distributional computation.
• Apply Kalman filtering to estimate the parameters of the one-dimensional Feller process intensity from high-frequency transaction data.
• Use the likelihood function derived from the analytical distribution to calibrate the model to observed point process data.

Experimental results

Research questions

• RQ1 What is the analytical form of the probability distribution function for a DSPP with an affine intensity process?
• RQ2 How can the affine structure of the intensity process be leveraged to compute exact likelihoods for point process models?
• RQ3 Can Kalman filtering effectively estimate the parameters of a Feller process intensity in high-frequency financial data?
• RQ4 What are the empirical implications of modeling intensity as an affine diffusion in financial transaction data?

Key findings

• The paper derives an exact analytical expression for the probability distribution function of the DSPP under affine intensity dynamics.
• The affine structure of the intensity process allows for closed-form computation of the moment-generating function of the integrated intensity.
• The model enables exact likelihood-based inference, which is critical for statistical estimation and model validation.
• Parameter estimation of the Feller process intensity is successfully performed using Kalman filtering on high-frequency transaction data.
• The framework provides a theoretically grounded and computationally feasible method for modeling financial point processes with stochastic intensity.