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The law of the supremum of a stable Lévy process with no negative jumps. (English) Zbl 1185.60051

Summary: Let \(X=(X_t)_{t\geq 0}\) be a stable Lévy process of index \(\alpha \in (1, 2)\) with no negative jumps and let \(S_t=\text{sup}_{0\leq s\leq t}X_s\) 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 \(\tau_x\).

MSC:

60G52 Stable stochastic processes
45D05 Volterra integral equations
60J75 Jump processes (MSC2010)
26A33 Fractional derivatives and integrals
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