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

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:$(β_{t},t\geq0)$: for any fixed $u>0$: \sinh(β_{u}) \stackrel{(law)}{=} \hatβ_{(\int^{u}_{0}ds\exp(2β_{s}))}. These identities in law for 2-dimensional processes allow to study the distributions of hitting times $T^θ_{c}\equiv\inf\{t:θ_{t} =c \}, (c>0)$, $T^θ_{-d,c}\equiv\inf\{t:θ_{t}\notin(-d,c) \}, (c,d>0)$ and more specifically of $T^θ_{-c,c}\equiv\inf\{t:θ_{t}\notin(-c,c) \}, (c>0)$ of the continuous winding processes $θ_{t}=\mathrm{Im}(\int^{t}_{0}\frac{dZ_{s}}{Z_{s}}), t\geq0$ of complex Ornstein-Uhlenbeck processes.