Marked doubly stochastic Poisson processes are a particular type of marked point processes which are characterized by thenumber of events in any time interval as being conditionallyPoisson distributed, given another positive stochastic processcalled intensity.Here we consider a subclass of these processes in which the intensity is assumed to be a deterministic function of another non-explosive 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 is here inits filtering, that is, in the computation of its conditional distribution, over a whole time interval, given an observed trajectory of realized events.Since in general this computation cannot be performed analytically, we propose a simulation method which provides an approximate solution, that relies on the reversible jump Markov chain Monte Carlo algorithm.Interestingly, the proposed filtering algorithm also allows the set up of a likelihood-based procedure for the estimation of the parameters of the model based on stochastic versions of the EM algorithm.From an application point of view, the class of processes considered finds application in many areas, such as quantum electronics, insurance and finance.Referring to the financial context, these processes are increasingly being used for the modeling of ultra-high-frequency data, where all trades or quotations changes, together with the corresponding time stamps, are recorded, to describe the intraday dynamics of the price of a financial asset.In this case, the form investigated for the intensity suggests an explicit interpretation in terms of the effect of the information release on the changes in price volatility and trading activity.The potential of the filtering and estimation methods proposed are illustrated through some simulation experiments as well as on a financial ultra-high-frequency data set of intraday S&P 500 future prices.
A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes
CENTANNI, Silvia;MINOZZO, Marco
2006-01-01
Abstract
Marked doubly stochastic Poisson processes are a particular type of marked point processes which are characterized by thenumber of events in any time interval as being conditionallyPoisson distributed, given another positive stochastic processcalled intensity.Here we consider a subclass of these processes in which the intensity is assumed to be a deterministic function of another non-explosive 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 is here inits filtering, that is, in the computation of its conditional distribution, over a whole time interval, given an observed trajectory of realized events.Since in general this computation cannot be performed analytically, we propose a simulation method which provides an approximate solution, that relies on the reversible jump Markov chain Monte Carlo algorithm.Interestingly, the proposed filtering algorithm also allows the set up of a likelihood-based procedure for the estimation of the parameters of the model based on stochastic versions of the EM algorithm.From an application point of view, the class of processes considered finds application in many areas, such as quantum electronics, insurance and finance.Referring to the financial context, these processes are increasingly being used for the modeling of ultra-high-frequency data, where all trades or quotations changes, together with the corresponding time stamps, are recorded, to describe the intraday dynamics of the price of a financial asset.In this case, the form investigated for the intensity suggests an explicit interpretation in terms of the effect of the information release on the changes in price volatility and trading activity.The potential of the filtering and estimation methods proposed are illustrated through some simulation experiments as well as on a financial ultra-high-frequency data set of intraday S&P 500 future prices.
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