Adaptive and robust nonparametric Bayesian contraction rates for discretely observed compound Poisson processes
Compound Poisson processes (CPPs) are the textbook example of pure jump stochastic processes, and they approximate arbitrarily well much richer classes of processes such as Lévy processes. They are characterised by the so-called Lévy jump distribution, N, driving the frequency at which jumps (randomly) occur and their (random) sizes. Hence, they provide a simple, yet fundamental, model for random shocks in a system applied in a myriad of problems within natural sciences, engineering and economics. In most applications, the underlying CPP is not perfectly observed: only discrete observations over a finite-time interval are available. Thus, the process may jump several times between two observations and we are effectively observing a random variable corrupted by a sum of a random number of copies of itself. Consequently, estimating N is a non-linear statistical inverse problem.
In the recent years, understanding the frequentist asymptotic behaviour of the Bayesian method in inverse
problems and, in particular, in this problem has received considerable attention. In this talk, we will present ongoing results on posterior contraction rates for the nonparametric density nu of N: we show two-sided stability estimates that guarantee that the classical theory in Ghosal, Ghosh, van der Vaart (2000) can be transferred to our problem, allowing us to use mixture and Gaussian priors for nu multidimensional; furthermore, the rates are robust to the observation interval, i.e. optimal adaptive inference can be made without specification of whether the regime is of high- or low-frequency; and, lastly, we propose an efficient infty-MCMC procedure to draw from the posterior for infinite dimensional priors. Given the diversity of the CCIMI members, we will attempt to introduce all these concepts during the presentation.