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Cut-off and hitting times of a sample of Ornstein-Uhlenbeck processes and its average

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

B. Lachaud
B. Lachaud*
Affiliation:
Université René Descartes - Paris 5
*
∗Postal address: MAP5, UMR CNRS 8145, Université Paris 5, 45 rue des Saints-Pères, 75270 Paris Cedex 06, France. Email address: beatrice.lachaud@math-info.univ-paris5.fr
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Abstract

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A cut-off phenomenon is shown to occur in a sample of n independent, identically distributed Ornstein-Uhlenbeck processes and its average. Their distributions stay far from equilibrium before a certain O(log(n)) time, and converge exponentially fast after. Precise estimates show that the total variation distance drops from almost 1 to almost 0 over an interval of time of length O(1) around log(n)/(2α), where α is the viscosity coefficient of the sampled process. The distribution of the hitting time of 0 by the average of the sample is computed. As n tends to infinity, the hitting time becomes concentrated around the cut-off instant, and its tails match the estimates given for the total variation distance.

Keywords

Cut-offhitting timeOrnstein-Uhlenbeck process

MSC classification

Primary: 60J60: Diffusion processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 4 , December 2005 , pp. 1069 - 1080
Copyright
© Applied Probability Trust 2005 

References

Alili, L., Patie, P. and Pedersen, J. (2005). Representations of the first hitting time of an Ornstein–Uhlenbeck process. To appear in Stoch. Models.Google Scholar
Barrera, J., Lachaud, B. and Ycart, B. (2005). Cutoff for n-tuples of exponentially converging processes. Submitted.Google Scholar
Bharucha-Reid, A. (1960). Elements of The Theory of Stochastic Processes and Their Applications. McGraw-Hill, New York.Google Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, New York.Google Scholar
Diaconis, P. (1996). The cut-off phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. USA 93, 1659–1664.CrossRefGoogle Scholar
Diaconis, P., Graham, R. and Morrison, J. (1990). Asymptotic analysis of a random walk on a hypercube with many dimensions. Random Structures Algorithms 1, 51–72.CrossRefGoogle Scholar
Feller, W. (1968). An Introduction to Probability Theory and its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Jeanblanc, M. and Rutkowski, M. (2000). Modelling of default risk: an overview. In Modern Mathematical Finance: Theory and Practice, Higher Education Press, Beijing, pp. 171–269.Google Scholar
Lansky, P., Sacerdote, L. and Tomasetti, F. (1995). On the comparison of Feller and Ornstein–Uhlenbeck models for neural activity. Biol. Cybernetics 73, 457–465.Google Scholar
Leblanc, B. and Scaillet, O. (1998). Path dependent options on yields in the affine term structure. Finance Stoch. 2, 349–367.Google Scholar
Pitman, J. and Yor, M. (1981). Bessel processes and infinitely divisible laws. In Stochastic Integrals (Proc. Symp. Univ. Durham, 1980; Lecture Notes Math. 851), Springer, Berlin, pp. 285–370.Google Scholar
Pollard, D. (2001). A User's Guide to Measure Theoretic Probability. Cambridge University Press.CrossRefGoogle Scholar
Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996; Lecture Notes Math. 1665), Springer, Berlin, pp. 301–413.Google Scholar
Saloff-Coste, L. (2004). Random walks on finite groups. In Probability on Discrete Structures (Encyclopaedia Math. Sci. 110), ed. Kesten, H., Springer, Berlin, pp. 263–346.CrossRefGoogle Scholar
Uhlenbeck, G. and Ornstein, L. (1930). On the theory of Brownian motion. Phys. Rev. 36, 823–841.CrossRefGoogle Scholar
Ycart, B. (1999). Cutoff for samples of Markov chains. ESAIM Prob. Statist. 3, 89–107.CrossRefGoogle Scholar
Ycart, B. (2001). Cutoff for Markov chains: some examples and applications. In Complex Systems, eds Goles, E. and Martı´nez, S., Kluwer, Dordrecht, pp. 261–300.CrossRefGoogle Scholar