On exit times of Levy-driven Ornstein–Uhlenbeck processes

We prove two martingale identities which involve exit times of Levy-driven Ornstein-Uhlenbeck processes. Using these identities we find an explicit formula for the Laplace transform of the exit time under the assumption that positive jumps of the Levy process are exponentially distributed.

[1]  Offer Kella,et al.  On hitting times for compound Poisson dams with exponential jumps and linear release rate , 2001, Journal of Applied Probability.

[2]  Martingales and first-exit times for the Ornstein-Uhlenbeck process with jumps , 2003 .

[3]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn = ρ X n–1 + Bn , 1980, Advances in Applied Probability.

[4]  Alexander Novikov,et al.  On the First Passage Time of an Autoregressive Process over a Level and an Application to a “Disorder” Problem , 1991 .

[5]  Aleksandr Alekseevich Borovkov,et al.  Stochastic processes in queueing theory , 1976 .

[6]  S. Wolfe.  On a continuous analogue of the stochastic difference equation Xn=ρXn-1+Bn , 1982 .

[7]  Shunji Osaki,et al.  On a first-passage problem for a cumulative process with exponential decay , 1976 .

[8]  Martin Jacobsen,et al.  Exit times for a class of piecewise exponential Markov processes with two-sided jumps , 2007 .

[9]  D. Darling,et al.  THE FIRST PASSAGE PROBLEM FOR A CONTINUOUS MARKOFF PROCESS , 1953 .

[10]  R. Melchers,et al.  First Passage Time of Filtered Poisson Process with Exponential Shape Function , 2003 .

[11]  F. Olver.  Asymptotics and Special Functions , 1974 .

[12]  Dimitar I. Hadjiev,et al.  The first passage problem for generalized Ornstein-Uhlenbeck processes with non-positive jumps , 1985 .

[13]  Alexander Novikov,et al.  On a piece-wise deterministic Markov process model , 2001 .

[14]  David Perry,et al.  FIRST-EXIT TIMES FOR POISSON SHOT NOISE , 2001 .