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

Abstract

Let X=(X_t)_t≥0 be a stable Lévy process of index α∈(1, 2) with no negative jumps and let S_t=sup_0≤s≤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 τ_x of X into [x, ∞), this further translates into an explicit series representation for the density function of τ_x.