On Brownian Motion Approximation of Compound Poisson Processes with Applications to Threshold Models

Abstract

Compound Poisson processes (CPP) constitute a fundamental class of stochastic processes and a basic building block for more complex jump-diffusion processes such as the Lévy processes. However, unlike those of a Brownian motion (BM), distributions of functionals, e.g. maxima, passage time, argmin and others, of a CPP are often intractable. The first objective of this paper is to propose a new approximation of a CPP by a BM so as to facilitate closed-form expressions in concrete cases. Specifically, we approximate, in some sense, a sequence of two-sided CPPs by a two-sided BM with drift. The second objective is to illustrate the above approximation in applications, such as the construction of confidence intervals of threshold parameters in threshold models, which include the threshold regression (also called two-phase regression or segmentation) and numerous threshold time series models. We conduct numerical simulations to assess the performance of the proposed approximation. We illustrate the use of our approach with a real data set.