Multivariate L\'evy-type drift change detection and mortality modeling

TL;DR

This paper develops a Bayesian quickest detection method for multivariate Lévy processes with Gaussian and jump components, using optimal stopping theory and a generalized Shiryaev-Roberts statistic, with applications to mortality modeling.

Contribution

It introduces a novel Bayesian approach for multivariate Lévy process change detection, including a solution for general priors and a new detection statistic.

Findings

Derived a generator for the posterior probability in multivariate Lévy processes.

Formulated and solved a free-boundary problem for optimal stopping.

Applied the method to mortality data, detecting drift changes in joint mortality forces.

Abstract

In this paper we give a solution to the quickest drift change detection problem for a multivariate L\'evy process consisting of both continuous (Gaussian) and jump components in the Bayesian approach. We do it for a general 0-modified continuous prior distribution of the change point. Classically, our criterion of optimality is based on a probability of false alarm and an expected delay of the detection, which is then reformulated in terms of a posterior probability of the change point. We find a generator of the posterior probability, which in case of general prior distribution is inhomogeneous in time. The main solving technique uses the optimal stopping theory and is based on solving a certain free-boundary problem. We also construct a Generelized Shiryaev-Roberts statistic, which can be used for applications. The paper is supplemented by two examples, one of which is further used to analyze Polish life tables (after proper calibration) and detect the drift change in the correlated force of mortality of men and women jointly.