ON BROWNIAN MOTION APPROXIMATION OF COMPOUND POISSON PROCESSES WITH APPLICATIONS TO THRESHOLD MODELS

Dong Li, Shiqing Ling, Howell Tong and Guangren Yang
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Dong Li: Center for Statistical Science and Department of Industrial Engineering, Tsinghua University, China
Shiqing Ling: Department of Statistics, Hong Kong University of Science and Technology
Howell Tong: School of Mathematical Sciences, University of Electronic Science and Technology of China
Guangren Yang: Department of Statistics, School of Economics, Jinan University, China

Advances in Decision Sciences, 2019, vol. 23, issue 2, 164-191

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 L´evy 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.

Keywords: Brownian motion; compound Poisson process; TAR; TARMA; TCHARM; TDAR; TMA; threshold regression (search for similar items in EconPapers)
JEL-codes: D31 D63 (search for similar items in EconPapers)
Date: 2019
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