### Abstract

Let B={B_{t}}_{t≥0} be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional A_{t}=∫_{0}^{t}e^{2Bs}ds,t≥0. Starting from a simple observation of generalized inverse Gaussian distributions with particular sets of parameters, we show, with the help of a result by Matsumoto and Yor (2000), that, for every x∈R and for every positive and finite stopping time τ of the process {e^{−Bt}A_{t}}_{t≥0}, the following identity in law holds: e^{Bτ}sinhx+β(A_{τ}),Ce^{Bτ}coshx+β̂(A_{τ}),e^{−Bτ}A_{τ}=(d)sinh(x+B_{τ}),Ccosh(x+B_{τ}),e^{−Bτ}A_{τ}, which extends an identity due to Bougerol (1983) in several aspects. Here β={β(t)}_{t≥0} and β̂={β̂(t)}_{t≥0} are one-dimensional standard Brownian motions, C is a standard Cauchy random variable, and B, β, β̂ and C are independent. The derivation of the above identity provides another proof of Bougerol's identity in law; moreover, a similar reasoning also enables us to obtain another extension for the three-dimensional random variable e^{Bτ}sinhx+β(A_{τ}),e^{Bτ},A_{τ}. By using an argument relevant to the derivation of those results, some invariance formulae for the Cauchy random variable C involving an independent Rademacher random variable, are presented as well.

Original language | English |
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Pages (from-to) | 5999-6037 |

Number of pages | 39 |

Journal | Stochastic Processes and their Applications |

Volume | 130 |

Issue number | 10 |

DOIs | |

Publication status | Accepted/In press - 2020 |

### Keywords

- Bougerol's identity
- Brownian motion
- Cauchy random variable
- Exponential functional
- Generalized inverse Gaussian distribution

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics