On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity

Abstract: Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes $(Z_{t}=X_{t}+iY_{t},t\geq0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$(\beta_{t},t\geq0)$: for any fixed $u>0$: \sinh(\beta_{u}) \stackrel{(law)}{=} \hat{\beta}{(\int^{u}{0}ds\exp(2\beta_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^{\theta}{c}\equiv\inf{t:\theta{t} =c }, (c>0)$, $T^{\theta}{-d,c}\equiv\inf{t:\theta{t}\notin(-d,c) }, (c,d>0)$ and more specifically of $T^{\theta}{-c,c}\equiv\inf{t:\theta{t}\notin(-c,c) }, (c>0)$ of the continuous winding processes $\theta_{t}=\mathrm{Im}(\int^{t}{0}\frac{dZ{s}}{Z_{s}}), t\geq0$ of complex Ornstein-Uhlenbeck processes.