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http://dx.doi.org/10.4134/BKMS.b180974

BOUNDS FOR EXPONENTIAL MOMENTS OF BESSEL PROCESSES

Makasu, Cloud (Departments of Mathematics and Applied Mathematics University of the Western Cape)

Publication Information

Bulletin of the Korean Mathematical Society / v.56, no.5, 2019 , pp. 1211-1217 More about this Journal

Abstract

Let $$0$ < ${\alpha}$ < ${\infty}$$ be fixed, and let $X=(X_t)_{t{\geq}0}$ be a Bessel process with dimension $$0$ < ${\theta}{\leq}1$$ starting at $x{\geq}0$ . In this paper, it is proved that there are positive constants A and D depending only on ${\theta}$ and ${\alpha}$ such that $$$E_x${\exp}[{\alpha}\;\max_{0{\leq}t{\leq}{\tau}}\;X_t$ $]$ $${\leq}AE_x${\exp}[D_{\tau}$ $]$ $$$$$ for any stopping time ${\tau}$ of X. This inequality is also shown to be sharp.

Keywords

Bessel processes; comparison principle; optimal stopping problem; Young inequality;

Citations & Related Records

Reference

1	D. L. Burkholder, Exit times of Brownian motion, harmonic majorization, and Hardy spaces, Advances in Math. 26 (1977), no. 2, 182-205. https://doi.org/10.1016/0001-8708(77)90029-9 DOI
2	R. D. DeBlassie, Stopping times of Bessel processes, Ann. Probab. 15 (1987), no. 3, 1044-1051. https://projecteuclid.org/euclid.aop/1176992079 DOI
3	L. E. Dubins, L. A. Shepp, and A. N. Shiryaev, Optimal stopping rules and maximal inequalities for Bessel processes, Theory Probab. Appl. 38 (1993), no. 2, 226-261; translated from Teor. Veroyatnost. i Primenen. 38 (1993), no. 2, 288-330. https://doi.org/10.1137/1138024 DOI
4	S. E. Graversen and G. Peskir, Maximal inequalities for Bessel processes, J. Inequal. Appl. 2 (1998), no. 2, 99-119. https://doi.org/10.1155/S102558349800006X
5	N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam, 1981.
6	D. Lepingle, Une remarque sur les lois de certains temps d'atteinte, in Seminar on Probability, XV (Univ. Strasbourg, Strasbourg, 1979/1980) (French), 669-670, Lecture Notes in Math., 850, Springer, Berlin, 1981.
7	D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Grundlehren der Mathematischen Wissenschaften, 293, Springer-Verlag, Berlin, 1991. https://doi.org/10.1007/978-3-662-21726-9 DOI
8	A. Yasue, Bessel-like processes and SDE, J. Math. Kyoto Univ. 44 (2004), no. 4, 799-809. https://doi.org/10.1215/kjm/1250281699 DOI
9	W. A. Rosenkrantz and S. Sawyer, $L^p$ estimates for stopping times of Bessel processes, Z. Wahrsch. Verw. Gebiete 41 (1977), no. 2, 145-151. https://doi.org/10.1007/BF00538418 DOI