Self-Exciting Point Processes: Identification and Control

This thesis addresses theoretical and practical aspects of identification and subsequent control of self-exciting point processes. The main contributions correspond to four separate scientific papers.In the first paper, we address the challenge of robust identification of controlled Hawkes processes in applications with sparsely available data. Specifically, we propose an alternative approach based on an expectation-maximization algorithm, which instrumentalizes the internal branching structure of the process, thus improving the estimator's convergence behavior. Additionally, we show that our method provides a tight lower bound for maximum-likelihood estimates. The relevance of the proposed technique is demonstrated on the practical application of credit collections and trading in the presence of macroeconomic news. The second and third paper focus on the optimal control of self-exciting point processes using the reinforcement-learning paradigm. Contrary to traditional reinforcement learning applications, environments driven by Hawkes-like dynamics feature an asynchronous action-reward relationship which complicates attributing actions to their consequent rewards, and thus hinders learning. To this end, we formulate a novel reward shaping theorem that provides a continuous reward analogue that enables learning in such environments. Furthermore, with the growing need for interpretable machine-learning models we formulate a monotonicity regularizer that embeds domain expertise into the learning. Our formulation overcomes the challenge of learning interpretable policies by constraining the policy space with a priori expected structural properties, producing state-feedback control laws that can be readily understood and implemented by human decision-makers. Again the results are developed in the context of credit collections but are straightforwardly applicable to other problems with self-exciting dynamics.Finally, the last paper consists of an empirical investigation of cryptocurrency market microstructure through the optics of Hawkes processes. We construct a 'reflexivity' index that measures the activity generated endogenously within cryptocurrency markets by fitting a univariate self-exciting Hawkes process with two classes of parametric kernels to high-frequency trading data. Our parsimonious model allows for an elegant separation and quantification of endogenous and exogenous dynamics, and thus allows for a direct market microstructure comparison with traditional asset classes in terms of identified branching ratios. Furthermore, we formulate a 'Hawkes disorder problem,' as a generalization of the established Poisson disorder problem, and provide a simulation-based approach to determining an optimal observation horizon---a critical consideration in the high-frequency finance context. Our analysis suggests that Bitcoin mid-price dynamics feature long-memory properties, well explained by the power-law kernel, at a level of criticality similar to fiat-currency markets.