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Continuous-time threshold AR(1) processes

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

O. Stramer ,
P. J. Brockwell  and
R. L. Tweedie
O. Stramer*
Affiliation:
University of Iowa
P. J. Brockwell*
Affiliation:
Royal Melbourne Institute of Technology
R. L. Tweedie*
Affiliation:
Colorado State University
*
∗ Postal Address: Department of Statistics and Actuarial Science, University of Iowa, Iowa City IA 52242, USA.
∗∗ Postal Address: Department of Mathematics, Royal Melbourne Institute of Technology, Melbourne 3001, Australia.
∗∗∗ Postal Address: Department of Statistics, Colorado State University, Fort Collins CO 80523, USA.
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Abstract

A threshold AR(1) process with boundary width 2δ > 0 was defined by Brockwell and Hyndman [5] in terms of the unique strong solution of a stochastic differential equation whose coefficients are piecewise linear and Lipschitz. The positive boundary-width is a convenient mathematical device to smooth out the coefficient changes at the boundary and hence to ensure the existence and uniqueness of the strong solution of the stochastic differential equation from which the process is derived. In this paper we give a direct definition of a threshold AR(1) process with δ = 0 in terms of the weak solution of a certain stochastic differential equation. Two characterizations of the distributions of the process are investigated. Both express the characteristic function of the transition probability distribution as an explicit functional of standard Brownian motion. It is shown that the joint distributions of this solution with δ = 0 are the weak limits as δ ↓ 0 of the distributions of the solution with δ > 0. The sense in which an approximating sequence of processes used by Brockwell and Hyndman [5] converges to this weak solution is also investigated. Some numerical examples illustrate the value of the latter approximation in comparison with the more direct representation of the process obtained from the Cameron–Martin–Girsanov formula and results of Engelbert and Schmidt [9]. We also derive the stationary distribution (under appropriate assumptions) and investigate stability of these processes.

Keywords

NON-LINEAR TIME SERIESCONTINUOUS-TIME AUTOREGRESSIONSTOCHASTIC DIFFERENTIAL EQUATIONAPPROXIMATION OF DIFFUSION PROCESSESRECURRENCESTATIONARY DISTRIBUTIONEXPONENTIAL ERGODICITYGAUSSIAN LIKELIHOOD

MSC classification

Primary: 60J60: Diffusion processes 60J35: Transition functions, generators and resolvents 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
Type
General Applied Probablity
Information
Advances in Applied Probability , Volume 28 , Issue 3 , September 1996 , pp. 728 - 746
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Work supported in part by NSF Grants DMS 9100392 and 9105745.

References

[1] Atkinson, J. D. and Caughey, T. K. (1968) Spectral density of piecewise linear first order systems excited by white noise. J. Non-Linear Mech. 3, 137–156.Google Scholar
[2] Azéma, J., Kaplan-Duflo, M. and Revuz, D. (1967) Mesure invariante sur les classes récurrentes des processus de Markov. Z. Wahrscheinlichkeitsth. 8, 157–181.CrossRefGoogle Scholar
[3] Bass, R. F. and Pardoux, E. (1987) Uniqueness for diffusions with piecewise constant coefficients. Prob. Theory Rel. Fields 76, 557–572.Google Scholar
[4] Billingsley, P. (1986) Probability and Measure. Wiley, New York.Google Scholar
[5] Brockwell, P. J. (1994) On continuous time threshold ARMA processes. J. Statist. Planning Inf. 39, 291–304.CrossRefGoogle Scholar
[6] Brockwell, P. J. and Hyndman, R. J. (1992) On continuous time threshold autoregression. Int. J. Forecasting 8, 157–173.Google Scholar
[7] Brockwell, P. J., Hyndman, R. J. and Grunwald, G. K. (1991) Continuous time threshold autoregressive models. Statist. Sinica 1, 401–410.Google Scholar
[8] Brockwell, P. J. and Stramer, O. (1995) On the approximation of continuous time threshold ARMA processes. Ann. Inst. Statist. Math. 47, 1–20.Google Scholar
[9] Engelbert, H. J. and Schmidt, W. (1984) On One-Dimensional Stochastic Differential Equations with Generalized Drift (Lecture Notes in Control and Information Sciences 69) . Springer, Berlin. pp. 143–155.Google Scholar
[10] Getoor, R. K. (1979) Transcience and recurrence of Markov processes. In Séminaire de Probabilités XIV. ed. Azéma, J. and Yor, M.. Springer, Berlin. pp. 397–409.Google Scholar
[11] Gikhman, I. I. and Skorokhod, A. V. (1969) Introduction to the Theory of Random Processes. Saunders, Philadelphia.Google Scholar
[12] Jones, R. H. (1981) Fitting a continuous time autoregression to discrete data. Appl. Time Series Anal. 2, 651–682.Google Scholar
[13] Karatzas, I. and Shreve, S. E. (1991) Brownian Motion and Stochastic Calculus. Springer, New York.Google Scholar
[14] Meyn, S. P. and Tweedie, R. L. (1993) Stability of Markovian processes II: Continuous time processes and sampled chains. Adv. Appl. Prob. 25, 487–517.Google Scholar
[15] Meyn, S. P. and Tweedie, R. L. (1993) Stability of Markovian processes III: Foster–Lyapunov criteria for continuous time processes. Adv. Appl. Prob. 25, 518–548.CrossRefGoogle Scholar
[16] Pollar, M. and Seigmund, D. (1985) A diffusion process and its applications to detecting a change in the drift of Brownian motion. Biometrika 72, 267–280.Google Scholar
[17] Stramer, O. and Tweedie, R. L. (1994) Stability and instability of continuous time Markov processes. In Probability, Statistics and Optimization: a Tribute to Peter Whittle. Vol. 12. pp. 173–184.Google Scholar
[18] Stramer, O., Tweedie, R. L. and Brockwell, P. J. (1996) Existence and stability of continuous time threshold ARMA processes. Statist. Sinica (accepted).Google Scholar
[19] Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer, Berlin.Google Scholar
[20] Tong, H. (1990) Non-linear Time Series: A Dynamical System Approach. Oxford University Press, Oxford.Google Scholar
[21] Tweedie, R. L. (1994) Topological conditions enabling use of Harris methods in discrete and continuous time. Acta Appl. Math. 34, 175–188.CrossRefGoogle Scholar