Bulletin of the
Korean Mathematical Society BKMS

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.

Keywords: Bessel processes, comparison principle, optimal stopping problem, Young inequality

MSC numbers: Primary 60G40, 34A40, 60E15