Joint law of the hitting time, overshoot and undershoot for a Lévy process

2016/03/08 by Laure Coutin, Coutin, Laure, Waly Ngom +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1603.02506

arxiv created 2016/03/08 · arxiv updated 2016/03/09

Abstract

Let be (X_t, t≥ 0) be a Lévy process which is the sum of a Brownian motion with drift and a compound Poisson process. We consider the first passage time τ_x at a fixed level x>0 by (X_t, t≥ 0) and K_x:= X_τx-x the overshoot and L_x:= x-X_τx^- the undershoot. We first study the regularity of the density of the first passage time. Secondly, we calculate the joint law of (τ_x, K_x, L_x).

Related

• Levy processes: Hitting time, overshoot and undershoot II - Asymptotic behaviour
• Using Differential Equations to Obtain Joint Moments of First-Passage Times of Increasing Levy Processes
• Levy Processes: Hitting time, overshoot and undershoot - part I: Functional equations
• Buffer-overflows: joint limit laws of undershoots and overshoots of reflected processes
• First passage time law for some Lévy processes with compound Poisson: Existence of a conditional density with incomplete observation