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.
Type de document :
Pré-publication, Document de travail
2010
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https://hal.archives-ouvertes.fr/hal-00505999
Contributeur : Stavros Vakeroudis <>
Soumis le : mardi 31 mai 2011 - 10:39:45
Dernière modification le : lundi 29 mai 2017 - 14:25:04
Document(s) archivé(s) le : dimanche 4 décembre 2016 - 21:23:39

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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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