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Level-crossing probabilities and first-passage times for linear processes

Part of: Stochastic processes Fredholm integral equations Mathematical economics Probabilistic methods, simulation and stochastic differential equations Inference from stochastic processes Markov processes

Published online by Cambridge University Press:  01 July 2016

Gopal K. Basak  and
Kwok-Wah Remus Ho
Gopal K. Basak*
Affiliation:
University of Bristol
Kwok-Wah Remus Ho*
Affiliation:
Hong Kong University of Science and Technology
*
∗ Postal address: Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK. Email address: magkb@bris.ac.uk
∗∗ Postal address: Department of Information and Systems Management, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. Email address: imremus@ust.hk
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Abstract

Discrete time-series models are commonly used to represent economic and physical data. In decision making and system control, the first-passage time and level-crossing probabilities of these processes against certain threshold levels are important quantities. In this paper, we apply an integral-equation approach together with the state-space representations of time-series models to evaluate level-crossing probabilities for the AR(p) and ARMA(1,1) models and the mean first passage time for AR(p) processes. We also extend Novikov's martingale approach to ARMA(p,q) processes. Numerical schemes are used to solve the integral equations for specific examples.

Keywords

First passage timelevel crossing probabilityintegral equationmartingaleAR processARMA process

MSC classification

Primary: 62M10: Time series, auto-correlation, regression, etc. 91B84: Economic time series analysis
Secondary: 45B05: Fredholm integral equations 60G17: Sample path properties 60J05: Discrete-time Markov processes on general state spaces 65C50: Other computational problems in probability
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 643 - 666
Copyright
Copyright © Applied Probability Trust 2004 

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References

[1] Atkinson, K. E. (1997). The Numerical Solution of Integral Equations of The Second Kind (Camb. Monogr. Appl. Comput. Math. 4). Cambridge University Press.Google Scholar
[2] Botta, R. F. and Horris, C. M. (1986). Approximation with generalized hyperexponential distributions: weak convergence results. Queueing Systems 1, 169–190.Google Scholar
[3] Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods, 2nd edn. Springer, New York.Google Scholar
[4] Chan, N. H. and Palma, W. (1998). State space modeling of long-memory processes. Ann. Statist. 26, 719–740.Google Scholar
[5] Ferrar, W. L. (1958). Integral Calculus. Oxford University Press.Google Scholar
[6] Greenberg, I. (1997). Markov chain approximation methods in a class of level-crossing problems. Operat. Res. Lett. 21, 153–158.CrossRefGoogle Scholar
[7] Novikov, A. A. (1990). On the first exit time of an autoregressive process beyond a level and an application to the ‘change point’ problem. Theory Prob. Appl. 35, 269–279.Google Scholar