The area of a spectrally positive stable process stopped at zero

Julien Letemplier (1) , Thomas Simon (1, 2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques

An identity in law for the area of a spectrally positive Lévy stable process stopped at zero is established. Extending that of Lefebvre for Brownian motion, it involves an inverse Beta random variable and the square of a positive stable random variable. This identity entails that the stopped area is distributed as the perpetuity of a spectrally negative Lévy process, and is hence self-decomposable. We also derive a convergent series representation for the density, whose behaviour at zero is shown to be Fréchet-like.

Mots clés: Exponential functional; Hitting time; Integrated process; Moments of Gamma type; Self-decomposability; Series representation; Stable Lévy process

Domaines: Probabilités [math.PR]

Julien Letemplier, Thomas Simon. The area of a spectrally positive stable process stopped at zero. 2014. ⟨hal-01070220⟩