Bounds for exponential moments of Bessel processes
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $0<\alpha<\infty$ be fixed, and let $X=(X_t)_{t\geq0}$ be a Bessel process with dimension $0<\theta\leq1$ starting at $x\geq0$. In this paper, it is proved that there are positive constants $A$ and $D$ depending only on $\theta$ and $\alpha$ such that \begin{equation*} \mathbf{E}_x\Biggl(\exp\bigl[\alpha\max\limits_{0\leq t\leq\tau}X_t\bigr]\Biggr)\leq A\mathbf{E}_x\Biggl(\exp[D\tau]\Biggr) \end{equation*} for any stopping time $\tau$ of $X$. This inequality is also shown to be sharp.
Abstract
Let $0<\alpha<\infty$ be fixed, and let $X=(X_t)_{t\geq0}$ be a Bessel process with dimension $0<\theta\leq1$ starting at $x\geq0$. In this paper, it is proved that there are positive constants $A$ and $D$ depending only on $\theta$ and $\alpha$ such that \begin{equation*} \mathbf{E}_x\Biggl(\exp\bigl[\alpha\max\limits_{0\leq t\leq\tau}X_t\bigr]\Biggr)\leq A\mathbf{E}_x\Biggl(\exp[D\tau]\Biggr) \end{equation*} for any stopping time $\tau$ of $X$. This inequality is also shown to be sharp.
참고문헌 (9)
- D.L Burkholder, "Exit times of Brownian motion, harmonic majorization, and Hardy spaces", Advances in Mathematics, 26(2), pp.182~205 (1977), Elsevier BV[DOI]
- R. D. DeBlassie, "Stopping times of Bessel processes", Ann. Probab, 15(3), pp.1044~1051 (1987)[DOI]
- L. E. Dubins, "Optimal Stopping Rules and Maximal Inequalities for Bessel Processes", Theory of Probability & Its Applications, 38(2), pp.226~261 (1994), Society for Industrial & Applied Mathematics (SIAM)[DOI]
- SE Graversen, "Maximal inequalities for bessel processes", Journal of Inequalities and Applications, 1998(2), pp.621735~ (1998), Springer Science and Business Media LLC[DOI]
- N. Ikeda, "Stochastic Dierential Equations and Diusion Processes" (1981), North-Holland Publishing Co.
- D. Lepingle, "Seminar on Probability, XV", pp.669~670 (1981), Springer
- D. Revuz, "Continuous Martingales and Brownian Motion" (1991), Springer-Verlag
- Walter A. Rosenkrantz, "L p estimates for stopping times of Bessel processes", Zeitschrift f�r Wahrscheinlichkeitstheorie und Verwandte Gebiete, 41(2), pp.145~151 (1977), Springer Science and Business Media LLC[DOI]
- A. Yasue, "Bessel-like processes and SDE", J. Math. Kyoto Univ, 44(4), pp.799~809 (2004)[DOI]