## Abstract

For a shot-noise process X(t) with Poisson arrival times and exponentially diminishing shocks of i.i.d. sizes, we consider the first time T_{b} at which a given level b > 0 is exceeded. An integral equation for the joint density of T_{b} and X(T_{b}) is derived and, for the case of exponential jumps, solved explicitly in terms of Laplace transforms (LTs). In the general case we determine the ordinary LT ψ_{b}(θ) = ∫_{0}^{∞} e-^{θt} P(T_{b} > t) dt of the function t → P(T_{b} > t) in terms of certain LTs derived from the distribution function H(x;t) = P(X(t) ≤ x), considered as a function of both variables x and t. Moreover, for G(t, u) = P(T_{b} > t, X(t) < u), that is the joint distribution function of sup_{0≤s≤t} X(s) and X(t), an integro-differential equation is presented, whose unique solution is G(t, u).

Original language | English |
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Pages (from-to) | 25-37 |

Number of pages | 13 |

Journal | Communications in Statistics. Part C: Stochastic Models |

Volume | 17 |

Issue number | 1 |

DOIs | |

State | Published - 2001 |

## Keywords

- First-exit time
- Integral equation
- Laplace transform
- Poisson process
- Shot noise

## ASJC Scopus subject areas

- Modeling and Simulation