×

zbMATH — the first resource for mathematics

On exit times of Levy-driven Ornstein-Uhlenbeck processes. (English) Zbl 1159.60019
The authors prove two martingale identities which involve exit times of Lévy-driven Ornstein-Uhlenbeck processes. Using these identities an explicit formula is found for the Laplace transform of the exit time under the assumption that positive jumps of the Lévy process are exponentially distributed.

MSC:
60G40 Stopping times; optimal stopping problems; gambling theory
60G44 Martingales with continuous parameter
60G51 Processes with independent increments; Lévy processes
60J75 Jump processes (MSC2010)
60E10 Characteristic functions; other transforms
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Borovkov, A., ()
[2] Borovkov, K.; Novikov, A., On a piece-wise deterministic Markov process model, Statist. probab. lett., 53, 4, 421-428, (2001) · Zbl 0993.60071
[3] Darling, D.A.; Siegert, A.J.F., The first passage problem for a continuous Markov process, Ann. math. statist., 24, 624-639, (1953) · Zbl 0053.27301
[4] Hadjiev, D., The first passage problem for generalized Ornstein-Uhlenbeck processes with nonpositive jumps, (), 80-90
[5] Jacobsen, M., Jensen, A., 2006, Exit times for a class of piecewise exponential Markov processes with two-sided jumps. Dept. of Applied Mathematics and Statistics, University of Copenhagen. Preprint No 5 · Zbl 1125.60080
[6] Kella, O.; Stadje, W., On hitting times for compound Poisson dams with exponential jumps and linear release rate, J. appl. probab., 38, 3, 781-786, (2001) · Zbl 0994.60092
[7] Novikov, A.A., On the first exit time of an autoregressive process beyond a level and an application to the “change-point” problem, Theory probab. appl., 35, 2, 269-279, (1990), (1991) · Zbl 0723.60044
[8] Novikov, A.A.; Èrgashev, B.A., Limit theorems for the time of crossing a level by an autoregressive process, Trudy mat. inst. Steklov., 202, 209-233, (1993), (in Russian). (Translation in Proc. Steklov Inst. Math. 1994, no. 4 (202), 169-186)
[9] Novikov, A.A., Martingales and first-exit times for the ornstein – uhlenbeck process with jumps, Theory probab. appl., 48, 340-358, (2003)
[10] Novikov, A.A.; Melchers, R.E.; Shinjikashvili, E.; Kordzakhia, N., First passage time of filtered Poisson process with exponential shape function, Probab. eng. mech., 20, 1, 33-44, (2005)
[11] Olver, F.W.J., Asymptotics and special functions, (1997), A.K. Peters Wellesley, MA · Zbl 0303.41035
[12] Perry, D.; Stadje, W.; Zacks, S., First-exit times for Poisson shot noise, Stoch. models, 17, 1, 25-37, (2001) · Zbl 1009.60031
[13] Tsurui, A.; Osaki, Sh., On a first-passage problem for a cumulative process with exponential decay, Stochastic process. appl., 4, 1, 79-88, (1976) · Zbl 0318.60028
[14] Wolfe, S., On a continuous analogue of the stochastic differential equation \(X_n = \rho X_{n - 1} + B_n\), Stochastic process. appl., 12, 301-312, (1982)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.