Let $B=\{ B_{t}\} _{t\ge 0}$ be a one-dimensional standard Brownian motion, to which we associate the exponential additive functional $A_{t}=\int _{0}^{t}e^{2B_{s}}ds,\,t\ge 0$. 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--Yor (2000), that for every $x\in \mathbb{R}$ and for every finite stopping time $\tau $ of the process $\{ e^{-B_{t}}A_{t}\} _{t\ge 0}$, there holds the identity in law \begin{align*} \left( e^{B_{\tau}}\!\sinh x+\beta (A_{\tau }), \, Ce^{B_{\tau}}\!\cosh x+\hat{\beta}(A_{\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) \stackrel{(d)}{=} \left( \sinh (x+B_{\tau }), \, C\cosh (x+B_{\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) , \end{align*} which extends an identity due to Bougerol (1983) in several aspects. Here $\beta =\{ \beta (t)\} _{t\ge 0}$ and $\hat{\beta}=\{ \hat{\beta}(t)\} _{t\ge 0}$ are one-dimensional standard Brownian motions, $C$ is a standard Cauchy variable, and $B$, $\beta $, $\hat{\beta}$ and $C$ are independent. Using an argument relevant to derivation of the above identity, we also present some invariance formulae for Cauchy variable involving an independent Rademacher variable.

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关于布朗运动和柯西随机变量指数函数的定律中的一些恒等式

令 $B=\{ B_{t}\} _{t\ge 0}$ 是一个一维标准布朗运动，我们将指数加性函数 $A_{t}=\int _{0}^ 与它相关联{t}e^{2B_{s}}ds,\,t\ge 0$。从对具有特定参数集的广义逆高斯分布的简单观察开始，我们在 Matsumoto--Yor (2000) 的结果的帮助下表明，对于每个 $x\in \mathbb{R}$ 和对于每个过程的有限停止时间 $\tau $ $\{ e^{-B_{t}}A_{t}\} _{t\ge 0}$，存在律\begin{align*} \ left( e^{B_{\tau}}\!\sinh x+\beta (A_{\tau }), \, Ce^{B_{\tau}}\!\cosh x+\hat{\beta}(A_ {\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) \stackrel{(d)}{=} \left( \sinh (x+B_{\tau }), \, C\cosh (x+B_{\tau }), \, e^{-B_{\tau }}\!A_{\tau } \right) , \end{align*} 扩展了一个由于 Bougerol (1983) 在几个方面的认同。这里 $\beta =\{ \beta (t)\} _{t\ge 0}$ 和 $\hat{\beta}=\{ \hat{\beta}(t)\} _{t\ge 0 }$是一维标准布朗运动，$C$是标准柯西变量，$B$、$\beta $、$\hat{\beta}$和$C$是独立的。使用与推导上述恒等式相关的参数，我们还提出了一些涉及独立 Rademacher 变量的柯西变量的不变性公式。