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

Abstract

Marked 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.

Biased Online Parameter Inference for State-Space Models

“…Similar, but different, ideas have appeared in several articles including [2,6,11]. These ideas are considered in the context of hidden Markov models and partially observed point processes respectively.…”

“…These ideas are considered in the context of hidden Markov models and partially observed point processes respectively. The key differences of our work to [6,11] ( [2] is for maximum likelihood estimation (MLE)) are as follows. In the context of [6] we do not use a type of 'sequential MCMC', in that our approach can be used explicitly for online Bayesian parameter estimation.…”

“…The key differences of our work to [6,11] ( [2] is for maximum likelihood estimation (MLE)) are as follows. In the context of [6] we do not use a type of 'sequential MCMC', in that our approach can be used explicitly for online Bayesian parameter estimation. The approach of [11] is less general, where the particles are not updated with PMCMC, and one has a block length of 1.…”

Static-parameter estimation in piecewise deterministic processes using particle Gibbs samplers

“…Throughout, we use (conditional) systematic resampling and resample only when the e ective sample size falls below N ESS WD 0:8N . The moves that update jumps in the algorithm are those used in Centanni and Minozzo (2006b) (except that for Example I, jump sizes are always sampled from their full conditional posterior distributions).…”

Networks of $$\cdot /G/\infty $$ · / G / ∞ queues with shot-noise-driven arrival intensities

“…The shot-noise-driven Cox process is attractive since it has this property. Statistical methods that filter the unobservable intensity process, based on Markov Chain Monte Carlo (MCMC) techniques, have been developed; see [3] and references therein. By filtering, they refer to the estimation of the intensity process in a given time interval, given a realized arrival process.…”