Let X=(Xt)t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let St=sup0≤s≤t Xs denote its running supremum for t>0. We show that the density function ft of St 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 ft. Recalling the familiar relation between St and the first entry time τx of X into [x, ∞), this further translates into an explicit series representation for the density function of τx.
"The law of the supremum of a stable Lévy process with no negative jumps." Ann. Probab. 36 (5) 1777 - 1789, September 2008. https://doi.org/10.1214/07-AOP376