The rise of electronic order-driven financial markets has brought a profusion of new high-frequency data to study, with an opportunity to understand the price formation mechanism at the smallest timescales. The original motivation of this thesis is to find a stochastic process that provides an accurate statistical dynamic description of this new data.
A critical analysis of the literature reveals a dichotomy between two main sorts of model, Hawkes processes and continuous-time Markov chains, each having qualities that the other lacks. In particular, models of the former sort are successful at capturing excitation effects between different event types but fail to incorporate the state of the market. We resolve this dichotomy by introducing state-dependent Hawkes processes, an extension of Hawkes processes where events can now interact with an auxiliary state process. These new stochastic processes provide us with the first model that features both excitation effects and an explicit feedback loop between events and the state of the market. The application of this new model to high-quality data demonstrates that the excitation effects are indeed strongly state-dependent.
State-dependent Hawkes processes come however with theoretical challenges: under which conditions do they exist, are they unique and do not explode? To answer these questions, we view state-dependent Hawkes processes as ordinary point processes of higher dimension, which we then generalise to the class of hybrid marked point processes. This class provides a framework that unifies and extends the existing high-frequency models. Since hybrid marked point processes are defined implicitly via their intensity, one can address the above questions by studying instead a Poisson-driven stochastic differential equation (SDE). We are able to solve this SDE under general assumptions that dispense with the Lipchitz condition usually required in the literature, which yields, as a corollary, the existence and uniqueness of non-explosive state-dependent Hawkes processes.Open Acces
