摘要：Let $X$ denote a spectrally positive stable process of index $\alpha \in (1, 2)$ whose Lévy measure has density $c x^{-\alpha - 1}$, $x > 0$ and let $S = \displaystyle \sup_{0 \leq t \leq 1} X_t$. Doney (Stochastics 80, 2008, 151-155) proved that the density of $S$ say $s$ behaves as $s (x) \sim c x^{-\alpha - 1}$ as $x \to \infty$. The proof given was nearly four pages long. Here, we: i) give a shorter and a more general proof of the same result; ii) derive the first known closed form expressions for $s (x)$ and the corresponding cumulative distribution function; iii) derive the order of the remainder in the asymptotic expansion for $s (x)$.
关键词：Asymptotic behavior; Stable process; Wright generalized hypergeometric $\Psi$ function
