Levy Processes: Hitting time, overshoot and undershoot - part I: Functional equations

Abstract

Let (X_t, t >=0) be a Levy process started at 0, with Levy measure nu, and T_x the first hitting time of level x>0: T_x := inf{t>=0; X_t>x}. Let F(theta,mu,rho,.) be the joint Laplace transform of (T_x, K_x, L_x): F(theta,mu,rho,x) := E (e^{-theta T_x - mu K_x - rho L_x} 1_{T_x<+infinity}), where theta>=0, mu>=0, rho>=0, x>0, K_x := X_{T_x} - x and L_x := x - X_{T_{x^-}}. If nu(R) < + \infinity and integral_1^{+\infty} e^{sy} nu (dy) < +infinity for some s>0, then we prove that F(theta,mu,rho,.) is the unique solution of an integral equation and has a subexponential decay at infinity when theta>0 or theta=0 and E(X_1)<0. If nu is not necessarily a finite measure but verifies integral_{-infinity}^{-1} e^{-sy} nu (dy) < +infinity for any s>0, then the x-Laplace transform of F(theta,mu,rho,.) satisfies some kind of integral equation. This allows us to prove that F(theta,mu,rho,.) is a solution to a second integral equation.