Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading

TL;DR

This paper develops a comprehensive mathematical framework for a generalized, non-Markovian Hawkes process with time-dependent baseline intensity and random jump sizes, and applies it to model trade arrivals in dark pools.

Contribution

It derives closed-form formulas for the Laplace transform, moments, and distribution of a generalized Hawkes process, extending existing models and enabling practical financial applications.

Findings

Closed-form Laplace transform, moments, and distribution derived.

Applied to model dark pool trade arrivals and analyze performance metrics.

Enhanced understanding of self-exciting point processes in finance.

Abstract

Hawkes processes are a class of simple point processes that are self-exciting and have clustering effect, with wide applications in finance, social networks and many other fields. This paper considers a self-exciting Hawkes process where the baseline intensity is time-dependent, the exciting function is a general function and the jump sizes of the intensity process are independent and identically distributed non-negative random variables. This Hawkes model is non-Markovian in general. We obtain closed-form formulas for the Laplace transform, moments and the distribution of the Hawkes process. To illustrate the applications of our results, we use the Hawkes process to model the clustered arrival of trades in a dark pool and analyze various performance metrics including time-to-first-fill, time-to-complete-fill and the expected fill rate of a resting dark order.