## Abstract

We give a representation of the solution for a stochastic linear equation of the form X_{t} = Y_{t} + ∫_{(0, t]} X _{s}- dZ_{s} where Z is a càdlàg semimartingale and Y is a càdlàg adapted process with bounded variation on finite intervals. As an application we study the case where Y and -Z are nondecreasing, jointly have stationary increments and the jumps of -Z are bounded by 1. Special cases of this process are shot-noise processes, growth collapse (additive increase, multiplicative decrease) processes and clearing processes. When Y and Z are, in addition, independent Lévy processes, the resulting X is called a generalized Ornstein-Uhlenbeck process.

Original language | American English |
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Pages (from-to) | 367-381 |

Number of pages | 15 |

Journal | Annals of Applied Probability |

Volume | 20 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2010 |

## Keywords

- Generalized Ornstein-Uhlenbeck process
- Growth collapse process
- Linear stochastic equation
- Risk process
- Shot-noise process