Discretization of Self-Exciting Peaks Over Threshold Models

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. 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.