First passage time law for some jump-diffusion processes : existence of a density

Let (Xt, t >= 0) be a diffusion process with jumps, sum of a Brownian motion with drift and a compound Poisson process. We consider T_x the first hitting time of a fixed level x > 0 by (Xt, t >= 0). We prove that the law of T_x has a density (defective when E(X1) < 0) with respect to the Lebesgue measure.

[1]  Robert C. Dalang,et al.  The law of the supremum of a stable Lévy process with no negative jumps , 2007, 0706.1503.

[2]  Hui Wang,et al.  First passage times of a jump diffusion process , 2003, Advances in Applied Probability.

[3]  V. Zolotarev.  The First Passage Time of a Level and the Behavior at Infinity for a Class of Processes with Independent Increments , 1964 .

[4]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[5]  P. Levy.  Processus stochastiques et mouvement brownien , 1948 .

[6]  A. E. Kyprianou,et al.  Overshoots and undershoots of Lèvy processes , 2006 .

[7]  Claude Lefèvre,et al.  The moments of ruin time in the classical risk model with discrete claim size distribution , 1998 .

[8]  Claude Lefèvre,et al.  The probability of ruin in finite time with discrete claim size distribution , 1997 .

[9]  A. Borovkov.  On the First Passage Time for One Class of Processes with Independent Increments , 1965 .

[10]  Stéphane Loisel,et al.  Finite Horizon Ruin Probabilities for Independent or Dependent Claim Amounts , 2007 .

[11]  B. Leblanc.  Modelisation de la volatilite d'un actif financier et applications , 1997 .

[12]  B. Roynette,et al.  Asymptotic behavior of the hitting time, overshoot and undershoot for some Lévy processes , 2008 .

[13]  R. Schilling.  Financial Modelling with Jump Processes , 2005 .

[14]  Xiongzhi Chen.  Brownian Motion and Stochastic Calculus , 2008 .

[15]  P. Vallois,et al.  Level crossing times for certain processes without positive jumps , 1997 .

[16]  C. Lefèvre,et al.  On finite-time ruin probabilities for classical risk models , 2008 .

[17]  Christophette Blanchet-Scalliet.  Processus à sauts et risque de défaut , 2001 .