Discretization of Self-Exciting Peaks Over Threshold Models

TL;DR

This paper develops a framework connecting discrete self-exciting point processes with continuous Hawkes processes, providing new theoretical insights and applications in financial extremal event analysis.

Contribution

It establishes a weak convergence of RCINAR(∞) processes to marked Hawkes processes and offers a new perspective on point process analysis in extreme value theory.

Findings

RCINAR(∞) processes converge to marked Hawkes processes with increased observation frequency.

Necessary and sufficient conditions for stationarity of RCINAR(∞) are provided.

Theoretical justification for the SEPOT model in financial extremal event analysis.

Abstract

In this paper, a framework on a discrete observation of (marked) point processes under the high-frequency observation is developed. Based on this framework, we first clarify the relation between random coefficient integer-valued autoregressive process with infinite order (RCINAR($\infty$)) and i.i.d.-marked self-exciting process, known as marked Hawkes process. For this purpose, we show that the point process constructed of the sum of a RCINAR($\infty$) converge weakly to a marked Hawkes process. This limit theorem establish that RCINAR($\infty$) processes can be seen as a discretely observed marked Hawkes processes when the observation frequency increases and thus build a bridge between discrete-time series analysis and the analysis of continuous-time stochastic process and give a new perspective in the point process approach in extreme value theory. Second, we give a necessary and sufficient condition of the stationarity of RCINAR($\infty$) process and give its random coefficient autoregressive (RCAR) representation. Finally, as an application of our results, we establish a rigorous theoretical justification of self-exciting peaks over threshold (SEPOT) model, which is a well-known as a (marked) Hawkes process model for the empirical analysis of extremal events in financial econometrics and of which, however, the theoretical validity have rarely discussed. Simulation results of the asymptotic properties of RCINAR($\infty$) shows some interesting implications for statistical applications.