The asymptotic behavior of densities related to the supremum of a stable process

الأصل · EN

If X is a stable process of index α∈(0,2) whose Lévy measure has density cx⁻α⁻¹ on (0,∞), and S₁=₀<ₜ≤₁Xₜ, it is known that P(S₁>x) Aα⁻¹x⁻α as x→∞ and P(S₁≤ x) Bα⁻¹ρ⁻¹xαρ as x0. [Here ρ=P(X₁>0) and A and B are known constants.] It is also known that S₁ has a continuous density, m say. The main point of this note is to show that m(x) Ax⁻⁽α⁺¹⁾ as x→∞ and m(x) Bxαρ⁻¹ as x0. Similar results are obtained for related densities.