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

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

Title

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

Authors

  • 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)

Abstract

Compound Poisson processes (CPP) constitute a fundamental class of stochastic processes and abasic 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

Classification-JEL

D31, D63

Pages

164-191

https://doi.org/10.47654/v23y2019i2p164-191

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ISSN 2090-3359 (Print)
ISSN 2090-3367 (Online)

Asia University, Taiwan

Scientific and Business World

4.7
2023CiteScore
 
86th percentile
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Q2 in Scopus
CiteScore 2023 = 4.7
CiteScoreTracker 2024 = 8.5
SNIP 2023 = 0.799
SJR Quartile = Q1
SJR 2024 = 0.814
H-Index = 20

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