Lévy Processes: Hitting Time, Overshoot and Undershoot I - Functional Equations

Let (Xt, t ≥ 0) be a Lévy process started at 0, with Lévy measure ν, and Tx the first hitting time of level x > 0 : Tx := inf {t ≥ 0; Xt > x}. Let F (θ, μ, ρ, .) be the joint Laplace transform of (Tx,Kx, Lx) : F (θ, μ, ρ, x) := E ( e1l{Tx<+∞} ) , where θ ≥ 0, μ ≥ 0, ρ ≥ 0, x ≥ 0, Kx := XTx − x and Lx := x−XT x − . If ν(R) < +∞ and ∫ +∞ 1 eν(dy) < +∞ for some s > 0, then we prove that F (θ, μ, ρ, .) is the unique solution of an integral equation and has a subexponential decay at infinity when θ > 0 or θ = 0 and E(X1) < 0. If ν is not necessarily a finite measure but verifies ∫ −1 −∞ eν(dy) < +∞ for any s > 0, then the xLaplace transform of F (θ, μ, ρ, .) satisfies some kind of integral equation. This allows us to prove that F (θ, μ, ρ, .) is a solution to a second integral equation.