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 (Zt=Xt+iYt,t≥0) including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:(βt,t≥0): for any fixed u>0: (βu) (law)= β(∫u0ds(2βs)). These identities in law for 2-dimensional processes allow to study the distributions of hitting times Tθc∈f\t:θt =c \, (c>0), Tθ-d,c∈f\t:θt(-d,c) \, (c,d>0) and more specifically of Tθ-c,c∈f\t:θt(-c,c) \, (c>0) of the continuous winding processes θt=Im(∫t0dZsZs), t≥0 of complex Ornstein-Uhlenbeck processes.
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