A note on the supremum of a stable process

Abstract

If X is a spectrally positive stable process of index α ∈ (1, 2) whose Lévy measure has density on (0,∞), and it is known that as x → ∞. It is also known that S ₁ has a continuous density, s say. The point of this note is to show that as x → ∞.

Keywords:

• stable process
• supremum
• density
• asymptotic behaviour

AMS Subject Classification:

• 60J30
• 60F15