From the multi-terms urn model to the self-exciting negative binomial distribution and Hawkes processes

TL;DR

This paper introduces a new multi-term urn model linking it to Hawkes processes, demonstrating how it captures correlations and phase transitions, and showing its effectiveness in modeling financial default data.

Contribution

It establishes a novel urn process that relates to Hawkes processes, revealing phase transitions and improving empirical default modeling accuracy.

Findings

The urn model converges to the self-exciting negative binomial distribution.

The phase transition from steady to non-steady states is characterized.

The urn process outperforms Hawkes in default data analysis.

Abstract

This study considers a new multi-term urn process that has a correlation in the same term and temporal correlation. The objective is to clarify the relationship between the urn model and the Hawkes process. Correlation in the same term is represented by the P\'{o}lya urn model and the temporal correlation is incorporated by introducing the conditional initial condition. In the double-scaling limit of this urn process, the self-exciting negative binomial distribution (SE-NBD) process, which is a marked Hawkes process, is obtained. In the standard continuous limit, this process becomes the Hawkes process, which has no correlation in the same term. The difference is the variance of the intensity function in that the phase transition from the steady to the non-steady state can be observed. The critical point, at which the power law distribution is obtained, is the same for the Hawkes and the urn processes. These two processes are used to analyze empirical data of financial default to estimate the parameters of the model. For the default portfolio, the results produced by the urn process are superior to those obtained with the Hawkes process and confirm self-excitation.