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Hawkes processes are a class of simple point processes that are self-excitingand have clustering effect, with wide applications in finance, social networksand many other fields. This paper considers a self-exciting Hawkes processwhere the baseline intensity is time-dependent, the exciting function is ageneral function and the jump sizes of the intensity process are independentand identically distributed non-negative random variables. This Hawkes model isnon-Markovian in general. We obtain closed-form formulas for the Laplacetransform, moments and the distribution of the Hawkes process. To illustratethe applications of our results, we use the Hawkes process to model theclustered arrival of trades in a dark pool and analyze various performancemetrics including time-to-first-fill, time-to-complete-fill and the expectedfill rate of a resting dark order.

Transform Analysis for Hawkes Processes with Applications in Dark Pool Trading