# 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.
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https://hal.archives-ouvertes.fr/hal-00505999
Contributor : Stavros Vakeroudis <>
Submitted on : Tuesday, May 31, 2011 - 10:39:45 AM
Last modification on : Tuesday, November 19, 2019 - 10:03:49 AM
Long-term archiving on: Sunday, December 4, 2016 - 9:23:39 PM

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• HAL Id : hal-00505999, version 4
• ARXIV : 1007.4648

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Stavros Vakeroudis. On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity.. 2010. ⟨hal-00505999v4⟩

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