## Abstract

Let B(t),t∈R be a standard Brownian motion. Define a risk process R_{u}^{δ}(t)=e^{δt}(u+c∫_{0}^{t}e^{−δs}ds−σ∫_{0}^{t}e^{−δs}dB(s)),t≥0 where u≥0 is the initial reserve, δ≥0 is the force of interest, c>0 is the rate of premium and σ>0 is a volatility factor. In this contribution we obtain an approximation of the Parisian ruin probability K_{S}^{δ}(u,T_{u}):=P{inft∈[0,S]sups∈[t,t+T_{u}]Ruδ(s)<0},S≥0 as u→∞ where T_{u} is a bounded function. Further, we show that the Parisian ruin time of this risk process can be approximated by an exponential random variable. Our results are new even for the classical ruin probability and ruin time which correspond to T_{u}≡0 in the Parisian setting.

Original language | English |
---|---|

Pages (from-to) | 34-44 |

Number of pages | 11 |

Journal | Statistics and Probability Letters |

Volume | 120 |

DOIs | |

Publication status | Published - 1 Jan 2017 |

Externally published | Yes |

## Keywords

- Brownian motion
- Parisian ruin
- Ruin probability
- Ruin time