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

TL;DR

This paper establishes identities in law linking planar Ornstein-Uhlenbeck processes and Brownian motion, enabling the analysis of winding process hitting times through Bougerol's identity.

Contribution

It introduces new identities in law for 2D Ornstein-Uhlenbeck processes, extending Bougerol's identity, to study winding process hitting times.

Findings

Derived identities in law for complex Ornstein-Uhlenbeck processes.

Connected winding times of processes to Bougerol's identity.

Provided tools for analyzing hitting times of winding processes.

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.