×

High excursions of Bessel process and other processes of Bessel type. (English. Russian original) Zbl 1427.60063

Dokl. Math. 100, No. 1, 346-348 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 487, No. 3, 238-241 (2019).
Summary: A high excursion probability for the modulus of a Gaussian vector process with independent identically distributed components is evaluated. It is assumed that the components have means zero and variances reaching its absolute maximum at a single point of the considered time interval. An important example of such processes is the Bessel process.

MSC:

60G15 Gaussian processes
60G60 Random fields
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Göing-Jaeschke, A.; Yor, M., No article title, Bernoulli, 9, 313-349 (2003) · Zbl 1038.60079 · doi:10.3150/bj/1068128980
[2] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 2nd ed. (Springer, Berlin, 1994). · Zbl 0804.60001
[3] A. N. Shiryaev, Essentials of Stochastic Finance: Facts, Models, Theory (Fazis, Moscow, 1998; World Scientific, Singapore, 1999).
[4] Estrella, A., No article title, Econometric Theory, 19, 1128-1143 (2003) · Zbl 1441.62677 · doi:10.1017/S0266466603196107
[5] Kiefer, J., No article title, Ann. Math. Stat., 30, 420-447 (1959) · Zbl 0134.36707 · doi:10.1214/aoms/1177706261
[6] Gikhman, I. I., No article title, Theory Probab. Appl., 2, 369-373 (1957) · doi:10.1137/1102027
[7] Pitman, J.; Yor, M., No article title, Electron. J. Probab., 4, 35 (1999)
[8] Delong, D. M., No article title, Commun. Stat. Theory Methods A, 10, 2197-2213 (1981) · Zbl 0467.60070 · doi:10.1080/03610928108828182
[9] V. I. Piterbarg, Asymptotic Methods in the Theory of Gaussian Processes and Fields (Am. Math. Soc., Providence, 1996). · Zbl 0841.60024
[10] Shepp, L. A., No article title, Ann. Math. Stat., 42, 946-951 (1971) · Zbl 0216.21203 · doi:10.1214/aoms/1177693323
[11] Zhdanov, A. I.; Piterbarg, V. I., No article title, Theory Probab. Appl., 63, 1-21 (2018) · Zbl 1414.60024 · doi:10.1137/S0040585X97T988885
[12] Hashorva, E.; Ji, L., No article title, Extremes, 18, 37-64 (2015) · Zbl 1315.60042 · doi:10.1007/s10687-014-0201-1
[13] Liu, P.; Ji, L., No article title, Stochastic Processes Appl., 127, 497-525 (2017) · Zbl 1354.60057 · doi:10.1016/j.spa.2016.06.016
[14] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation (Cambridge Univ. Press, Cambridge, 1987). · Zbl 0617.26001 · doi:10.1017/CBO9780511721434
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.