A sequential Monte Carlo filter in a class of marked doubly stochastic Poisson processes

A central problem faced when modelling with doubly stochastic Poisson processes is the filtering of the underlying, and typically unobservable, intensity. In the financial context, the filtering of the intensity can be used in the numerical computation of a number of financial problems, such as price forecast, derivative pricing and hedging. This is a nonlinear filtering problem which does not admit an explicit solution, and which, in similar circumstances, is usually tackled by approximation or simulation techniques. It can be seen that, for a particular choice of the intensity, the filtering problem, which can be stated in terms of the conditional distribution of the whole intensity, in a given time interval, given a realized trajectory of the doubly stochastic Poisson process, can be solved by a simulation algorithm based on the reversible jump Markov chain Monte Carlo technique. On the other hand, here a time recursion is constructed, to characterize the filtering distribution at certain time instants in terms of past filtering distributions. To approximate the filter, an algorithm can be constructed, in which samples are drawn recursively from each filtering distribution. These samples can be viewed, in the optic of particle filters, as discrete approximations with random support (the particles) of the distributions of interest. From these approximate distributions, other quantities can be derived such as the conditional expectation of the intensity, given the observations, at any given time instant.