The law of the supremum of a stable Lévy process with no negative jumps

Details The law of the supremum of a stable Lévy process with no negative jumps The law of the supremum of a stable Lévy process with no negative jumps

2007-06-11

arxiv.org/abs/0706.1503v3

Annals of Probability 2008, Vol. 36, No. 5, 1777-1789

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AOP376 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/07-AOP376

Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be characterized as the unique solution to a weakly singular Volterra integral equation of the first kind or, equivalently, as the unique solution to a first-order Riemann--Liouville fractional differential equation satisfying a boundary condition at zero. This yields an explicit series representation for $f_t$. Recalling the familiar relation between $S_t$ and the first entry time $\tau_x$ of $X$ into $[x,\infty)$, this further translates into an explicit series representation for the density function of $\tau_x$.

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