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 $(Z_t=X_t+i{Y}_t,t\ge 0)$ including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:$({\beta }_t,t\ge 0)$: 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 $T_c^{\theta }\equiv \inf \{t:{\theta }_t=c\},(c>0)$, $T_{-d,c}^{\theta }\equiv \inf \{t:{\theta }_t\notin (-d,c)\},(c,d>0)$ and more specifically of $T_{-c,c}^{\theta }\equiv \inf \{t:{\theta }_t\notin (-c,c)\},(c>0)$ of the continuous winding processes ${\theta }_t=\mathrm{Im}({\int }_0^t\frac{d{Z}_s}{Z_s}),t\ge 0$ of complex Ornstein-Uhlenbeck processes.

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