×

zbMATH — the first resource for mathematics

Large deviations related to the law of the iterated logarithm for Itô diffusions. (English) Zbl 1434.60088
Summary: When a Brownian motion is scaled according to the law of the iterated logarithm, its supremum converges to one as time tends to zero. Upper large deviations of the supremum process can be quantified by writing the problem in terms of hitting times and applying a result of V. Strassen [in: Proceedings of the fifth Berkeley symposium on mathematical statistics and probability. Vol. II. Part 1. Berkeley, CA: University of California Press. 315–343 (1967; Zbl 0201.49903)] on hitting time densities. We extend this to a small-time large deviations principle for the supremum of scaled Itô diffusions, using as our main tool a refinement of Strassen’s result due to H. R. Lerche [Boundary crossing of Brownian motion. Its relation to the law of the iterated logarithm and to sequential analysis. Berlin: Springer (1986; Zbl 0604.62075)].
MSC:
60F10 Large deviations
60J60 Diffusion processes
60J65 Brownian motion
Software:
GAUSSIAN
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] Lucia Caramellino, Strassen’s law of the iterated logarithm for diffusion processes for small time, Stochastic Process. Appl. 74 (1998), no. 1, 1-19. · Zbl 0932.60034
[2] Amir Dembo and Ofer Zeitouni, Large deviations techniques and applications, second ed., Stochastic Modelling and Applied Probability, vol. 38, Springer-Verlag, New York, 1998. · Zbl 0896.60013
[3] Fuqing Gao and Shaochen Wang, Asymptotic behaviors for functionals of random dynamical systems, Stoch. Anal. Appl. 34 (2016), no. 2, 258-277. · Zbl 1344.60028
[4] Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, second ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. · Zbl 0734.60060
[5] Hans Rudolf Lerche, Boundary crossing of Brownian motion, Lecture Notes in Statistics, vol. 40, Springer-Verlag, Berlin, 1986. · Zbl 0604.62075
[6] Henry P. McKean, Jr., Stochastic integrals, Probability and Mathematical Statistics, No. 5, Academic Press, New York-London, 1969.
[7] Vladimir I. Piterbarg, Asymptotic methods in the theory of Gaussian processes and fields, Translations of Mathematical Monographs, vol. 148, American Mathematical Society, Providence, RI, 1996, Translated from the Russian by V. V. Piterbarg, Revised by the author.
[8] Volker Strassen, Almost sure behavior of sums of independent random variables and martingales, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Univ. California Press, Berkeley, Calif., 1967, pp. 315-343. · Zbl 0931.41017
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.