Perspectives on Hawkes Processes

This thesis addresses Hawkes point processes in seven scientific papers. We build theoretical bridges between Hawkes processes and other mathematical concepts—such as time series, branching random walks, or graph theory. In Paper A, we represent monotype Hawkes processes as limits of time-series based point processes. We examine the corresponding time series, the integer-valued autoregressive (INAR) time series of infinite order, in some detail. Furthermore, we point out structural analogies between Hawkes processes and INAR time series. In Paper B, we represent multitype Hawkes processes as type/space projections of certain branching random walks. This representation allows to generalize the convergence result from Paper A to the multitype case. Furthermore, it opens the door to generalizations of Hawkes processes that might be interesting in applications. In Paper C, we introduce a nonparametric estimation procedure for multitype Hawkes processes: we discretize Hawkes-process data. From Paper A and Paper B, we know that the resulting bin-count sequences can be approximated by INAR time series. Thus, we estimate the INAR parameters by standard methods and¡ retranslate the results into the point process world. In Paper D, we represent multitype Hawkes processes as directed weighted graphs. These ‘Hawkes graphs’ summarize the branching structure of a Hawkes process in a compact, yet meaningful way. We point out how the graphical perspective is also fertile mathematically, implementation-wise, and pedagogically. Furthermore, we apply the estimation method from Paper C to infer the Hawkes graph from large datasets. We pay special attention to computational issues. In Paper E, we apply the methods and concepts from Paper C and Paper D to limit-order-book data. In particular, we extend our estimation procedure to the marked case. The various estimation results allow insights into market microstructure. In Paper F, we give the results of a simulation study, where we compare our estimation procedure with maximum-likelihood estimation. Finally, in Paper G, we consider a certain critical case of the monotype Hawkes process. We study the critical Hawkes process by applying results from critical cluster fields, renewal theory, and regular variation. We discuss a possible Poisson embedding and a Palm version of the critical Hawkes process. Our methods give possible directions for the open discussion of multitype critical Hawkes processes as well as of critical INAR times series.