A new three-dimensional extension of Bougerol's identity in law (Q6597219)

From the text: ``To illustrate Bougerol's identity in law consider the following exponential functional with an associated Brownian motion \((B_s, s \ge 0)\) such that \N\[\N A_t = \int_0^t ds e^{2B_s}, t \ge 0.\N\]\N\textit{P. Bougerol}'s identity in law [Ann. Inst. Henri Poincaré, Nouv. Sér., Sect. B 19, 369--391 (1983; Zbl 0533.60010)] states that, for every fixed \(t\): \N\[\N\sinh(B_t) \overset{(\mathrm{law})}{=} \beta_{A_t} \N\]\Nwhere \(\beta\) is an independent, real Brownian motion. [\(\ldots\)] \N\NProofs [\(\ldots\)] in three dimensions are widely available through the less intuitive conventional methods of proof for the identity in law. We build upon \textit{S. Vakeroudis}'s new two-dimensional extension [Probab. Surv. 9, 411--437 (2012; Zbl 1278.60125)] in the third dimension, providing a more intuitive approach to foster higher dimensional expansion of this identity in law.''