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**On several two-boundary problems for a particular class of Lévy processes.**
*(English)*
Zbl 1138.60038

The authors study the joint law of the first exit time and the overshoot from an interval for a special class of Lévy processes; more specifically, the authors consider a compound Poisson process with exponentially distributed negative jumps and arbitrary positive jumps. The special structure of this Lévy processes allows them to derive closed form expressions for the integral transform of this joint law. In addition, the authors study the joint law of the entry into an interval and the value of the process at the entry time for the same class of processes.

Reviewer: Antonis Papapantoleon (Wien)

### MSC:

60G51 | Processes with independent increments; Lévy processes |

60J50 | Boundary theory for Markov processes |

### Keywords:

compound Poisson process with two sided jumps; first exit time; overshoot; first entry time; integral transforms### References:

[1] | Bertoin, J.: Lévy Processes. Cambridge University Press (1996) · Zbl 0861.60003 |

[2] | Bertoin, J.: Exponential decay and ergodicity of completely asymmetric Lévy process in a finite interval. Ann. Appl. Probab. 7, 156–169 (1997) · Zbl 0880.60077 · doi:10.1214/aoap/1034625257 |

[3] | Borovkov, A.A.: Stochastic Processes in Queueing Theory. Springer, Berlin (1976) · Zbl 0319.60057 |

[4] | Bratiychuk, N.S., Gusak, D.V.: Boundary Problems for Processes with Independent Increments. Naukova Dumka, Kiev (1990) (in Russian) · Zbl 0758.60074 |

[5] | Ditkin, V.A., Prudnikov, A.P.: Operational Calculus. Moscow (1966). Russian edition · Zbl 0126.31402 |

[6] | Emery, D.J.: Exit problem for a spectrally positive process. Adv. Appl. Probab. 5, 498–520 (1973) · Zbl 0297.60035 · doi:10.2307/1425831 |

[7] | Gihman, I.I., Skorokhod, A.V.: Theory of Stochastic Processes, vol. 2. Springer, Berlin (1975). Translated from the Russian by S. Kotz · Zbl 0305.60027 |

[8] | Kadankov, V.F., Kadankova, T.V.: On the distribution of duration of stay in an interval of the semi-continuous process with independent increments. Random Oper. Stoch. Equ. (ROSE) 12(4), 365–388 (2004) · Zbl 1119.60034 |

[9] | Kadankov, V.F., Kadankova, T.V.: On the distribution of the first exit time from an interval and the value of overshoot through the boundaries for processes with independent increments and random walks. Ukr. Math. J. 10(57), 1359–1384 (2005) · Zbl 1093.60020 |

[10] | Kadankova, T.V.: On the distribution of the number of the intersections of a fixed interval by the semi-continuous process with independent increments. Theor. Stoch. Process. 1–2, 73–81 (2003) · Zbl 1064.60100 |

[11] | Kadankova, T.V.: On the joint distribution of supremum, infimum and the magnitude of a process with independent increments. Theor. Probab. Math. Stat. 70, 54–62 (2004) · Zbl 1075.60093 |

[12] | Kyprianou, A.E.: A martingale review of some fluctuation theory for spectrally negative Lévy processes. Research report, Utrecht University (2003) |

[13] | Pecherskii, E.A.: Rogozin, B.A. (1969). On joint distributions of random variables associated with fluctuations of a process with independent increments. Theor. Probab. Appl. 14, 410–423 |

[14] | Petrovskii, I.G.: Lectures on the Theory of Integral Equations. Nauka, Moscow (1965). Russian edition |

[15] | Pistorius, M.R.: A potential theoretical review of some exit problems of spectrally negative Lévy processes. Séminaire Probab. 38, 30–41 (2004) |

[16] | Pistorius, M.R.: On exit and ergodicity of the spectrally negative Lévy process reflected in its infimum. J. Theor. Probab. 17, 183–220 (2004) · Zbl 1049.60042 · doi:10.1023/B:JOTP.0000020481.14371.37 |

[17] | Rogozin, B.A.: On distributions of functionals related to boundary problems for processes with independent increments. Theor. Probab. Appl. 11(4), 656–670 (1966) · Zbl 0178.52701 |

[18] | Suprun, V.N.: Ruin problem and the resolvent of a terminating process with independent increments. Ukr. Math. J. 28(1), 53–61 (1976) (English transl.) · Zbl 0349.60075 · doi:10.1007/BF01559226 |

[19] | Suprun, V.N., Shurenkov, V.M.: On the resolvent of a process with independent increments terminating at the moment when it hits the negative real semiaxis. In: Studies in the Theory of Stochastic Processes, pp. 170–174. Institute of Mathematics, Academy of Sciences of UKrSSR, Kiev (1976) |

[20] | Zolotarev, V.M.: The first passage time of a level and the behaviour at infinity for a class of processes with independent increments. Theor. Probab. Appl. 9(4), 653–664 (1964) · Zbl 0149.12903 · doi:10.1137/1109090 |

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