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

2010/07/27 by Stavros Vakeroudis, S. Vakeroudis, Vakeroudis, Stavros
Biochemistry, Genetics and Molecular Biology · Economics, Econometrics and Finance · Mathematics · #Diffusion and Search Dynamics #Point processes and geometric inequalities #Stochastic processes and financial applications #math.PR

paper · pdf · doi:10.48550/arxiv.1007.4648

arxiv created 2011/05/31 · arxiv updated 2011/06/01

Abstract

Some identities in law in terms of planar complex valued Ornstein-Uhlenbeck processes (Z_t=X_t+iY_t,t≥0) including planar Brownian motion are established and shown to be equivalent to the well known Bougerol identity for linear Brownian motion:(β_t,t≥0): for any fixed u>0: \sinh(β_u) \stackrel(law)= β_(∫^u₀dsexp(2β_s)). These identities in law for 2-dimensional processes allow to study the distributions of hitting times T^θ_c≡inf\t:θ_t =c \, (c>0), T^θ_-d,c≡inf\t:θ_t∉(-d,c) \, (c,d>0) and more specifically of T^θ_-c,c≡inf\t:θ_t∉(-c,c) \, (c>0) of the continuous winding processes θ_t=Im(∫^t₀\fracdZ_sZ_s), t≥0 of complex Ornstein-Uhlenbeck processes.

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