Nonparametric Bayesian inference for multidimensional compound Poisson processes

TL;DR

This paper develops a nonparametric Bayesian method for estimating the jump size density and intensity of multidimensional compound Poisson processes from discrete data, establishing optimal convergence rates.

Contribution

It introduces the first nonparametric Bayesian estimators for multidimensional Lévy processes and analyzes their convergence rates, a novel contribution to the field.

Findings

Posterior contraction rates are established and shown to be optimal.

Bayesian point estimates converge to true parameters at these rates.

The method is applicable to discretely observed multidimensional Lévy processes.

Abstract

Given a sample from a discretely observed multidimensional compound Poisson process, we study the problem of nonparametric estimation of its jump size density $r_0$ and intensity $\lambda_0$. We take a nonparametric Bayesian approach to the problem and determine posterior contraction rates in this context, which, under some assumptions, we argue to be optimal posterior contraction rates. In particular, our results imply the existence of Bayesian point estimates that converge to the true parameter pair $(r_0,\lambda_0)$ at these rates. To the best of our knowledge, construction of nonparametric density estimators for inference in the class of discretely observed multidimensional L\'{e}vy processes, and the study of their rates of convergence is a new contribution to the literature.