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

Stavros Vakeroudis (1, 2, 3)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 ENS-PSL - École normale supérieure - Paris
3 MODAL'X - Modélisation aléatoire de Paris X

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.

Mots clés: Planar Brownian motion; Ornstein-Uhlenbeck processes; continuous winding process; Bougerol's identity; exit time from a cone.; exit time from a cone

Domaines: Probabilités [math.PR]

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⟩