A Monte Carlo Approach to Filtering for a Class of Marked Doubly Stochastic Poisson Processes

Abstract

AbstractMarked doubly stochastic Poisson processes are a particular type of marked point processes that are characterized by the number of events in any time interval as being conditionally Poisson distributed, given another positive stochastic process called intensity. Here we consider a subclass of these processes in which the intensity is assumed to be a deterministic function of another nonexplosive marked point process. In particular, we will investigate an intensity jump process with an exponential decay having an analytic form for the distribution of the times and sizes of the jumps, which can be seen as a generalization of the classical shot noise process. Assuming that the intensity is unobservable, interest here is in its filtering, that is, in the computation of its conditional distribution, over a whole time interval, given an observed trajectory of realized events. Because, in general, this computation cannot be performed analytically, we propose a simulation method that provides an approximate solution, which relies on the reversible-jump Markov chain Monte Carlo algorithm. Interestingly, the proposed filtering algorithm also allows the setup of a likelihood-based procedure for the estimation of the parameters of the model based on stochastic versions of the expectation–maximization (EM) algorithm. The potential of the filtering and estimation methods proposed are illustrated through some simulation experiments as well as on a financial ultra-high-frequency dataset of intraday S&P500 futures prices.KEY WORDS: Cox processMarked point processMonte Carlo EMReversible-jump Markov chain Monte CarloShot noise processUltra-high-frequency data