On the weak-hash metric for boundedly finite integer-valued measures

It is known that the space of boundedly finite integer-valued measures on a complete separable metric space becomes itself a complete separable metric space when endowed with the weak-hash metric. It is also known that convergence under this topology can be characterised in a way that is similar to the weak convergence of totally finite measures. However, the original proofs of these two fundamental results assume that a certain term is monotonic, which is not the case as we give a counterexample. We manage to clarify these original proofs by addressing specifically the parts that rely on this assumption and finding alternative arguments.Comment: Minor typos corrected, Bulletin of the Australian Mathematical Society, 201

High-frequency financial data modelling with hybrid marked point processes

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

Hybrid marked point processes: characterisation, existence and uniqueness

We introduce a class of hybrid marked point processes, which encompasses and extends continuous-time Markov chains and Hawkes processes. While this flexible class amalgamates such existing processes, it also contains novel processes with complex dynamics. These processes are defined implicitly via their intensity and are endowed with a state process that interacts with past-dependent events. The key example we entertain is an extension of a Hawkes process, a state-dependent Hawkes process interacting with its state process. We show the existence and uniqueness of hybrid marked point processes under general assumptions, extending the results of Massouli\'e (1998) on interacting point processes.Comment: v6: introduction updated with reference to application of state-dependent Hawkes processe