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Large deviation estimates of the crossing probability for pinned Gaussian processes

Part of: Limit theorems Probabilistic methods, simulation and stochastic differential equations Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Lucia Caramellino  and
Barbara Pacchiarotti
Lucia Caramellino*
Affiliation:
Università di Roma Tor Vergata
Barbara Pacchiarotti*
Affiliation:
Università di Roma Tor Vergata
*
∗ Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy.
∗ Postal address: Dipartimento di Matematica, Università di Roma Tor Vergata, Via della Ricerca Scientifica, I-00133 Roma, Italy.
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Abstract

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The paper deals with the asymptotic behavior of the bridge of a Gaussian process conditioned to stay in n fixed points at n fixed past instants. In particular, functional large deviation results are stated for small time. Several examples are considered: integrated or not fractional Brownian motions and m-fold integrated Brownian motion. As an application, the asymptotic behavior of the exit probability is studied and used for the practical purpose of the numerical computation, via Monte Carlo methods, of the hitting probability up to a given time of the unpinned process.

Keywords

Conditioned Gaussian processreproducing kernel Hilbert spaceslarge deviationsexit time probabilityMonte Carlo method

MSC classification

Primary: 60F10: Large deviations 60G15: Gaussian processes
Secondary: 65C05: Monte Carlo methods
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 2 , June 2008 , pp. 424 - 453
Copyright
Copyright © Applied Probability Trust 2008 

References

Azencott, R. (1980). Grande Déviations et Applications (École d'été de Probabilités de Saint Flour VIII; Lecture Notes Math. 774). Springer, Berlin.Google Scholar
Baldi, P. and Caramellino, L. (2002). Asymptotics of hitting probabilities for general one-dimensional pinned diffusions. Ann. Appl. Prob. 12, 1071–1095.Google Scholar
Baldi, P. and Pacchiarotti, B. (2006). Explicit computation of second order moments of importance sampling estimators for fractional Brownian motion. Bernoulli 12, 663–688.Google Scholar
Baldi, P., Caramellino, L. and Iovino, M. G. (1999). Pricing general barrier options: a numerical approach using sharp large deviations. Math. Finance 9, 293–321.Google Scholar
Chen, X. and Li, W. V. (2003). Quadratic functionals and small ball probabilities for the m-fold integrated Brownian motion. Ann. Prob. 31, 1052–1077.Google Scholar
Cheridito, P. (2001). Mixed fractional Brownian motion. Bernoulli 7, 913–934.Google Scholar
Dembo, A. and Zeitouni, O. (1992). Large Deviations Techniques and Applications. Jones and Bartlett, Boston, MA.Google Scholar
Deuschel, J. D. and Stroock, D. W. (1989). Large Deviations. Academic Press, Boston, MA.Google Scholar
Galleani, L., Sacerdote, L., Tavella, P. and Zucca, C. (2003). A mathematical model for the atomic clock error. Metrologia 40, 257–264.Google Scholar
Gasbarra, D., Sottinen, T. and Valkeila, E. (2007). Gaussian bridges. In The Able Symposium 2005: Stochastic Analysis and Applications, eds Benth, F. et al., Springer, Berlin, pp. 361–382.Google Scholar
Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001). A canonical process for estimation of convex functions: the ‘invelope’ of integrated Brownian motion t 4 . Ann. Statist. 29, 1620–1652.Google Scholar
Mandjes, M., Mannersalo, P., Norros, I. and van Uitert, M. (2006). Large deviations of infinite intersections of events in Gaussian processes. Stoch. Process. Appl. 116, 1269–1293.Google Scholar
Molchan, G. and Khokhlov, A. (2004). Small values of the maximum or the integral of fractional Brownian motion. J. Statist. Phys. 114, 923–946.Google Scholar
Norros, I. and Saksman, A. (2007). Local independence of fractional Brownian motion. Preprint. Available at http://arXiv.org/abs/0711.4809.Google Scholar