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

S. Vakeroudis (1)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires

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 (beta(t), t >= 0): for any fixed u > 0, sinh beta(u) -(law) (beta) over cap (f0u) (ds exp (2 beta s)), with ((beta) over cap (t), t >= 0) a Brownian motion, independent of beta. These identities in law for two-dimensional processes allow us to study the distributions of hitting times T-c(theta) equivalent to inf{t : theta(t) = c} (c > 0), T--d, c(theta) equivalent to inf{t : theta(t) is not an element of (-d, c)} (c, d > 0) and more specifically of T--c, c(theta) equivalent to inf{t : theta(t) is not an element of (-c, c)}, (c > 0) of the continuous winding processes theta(t) = Im(f(0)(t) Z(s)(-1) dZ(s)), t >= 0, of complex valued Ornstein-Uhlenbeck processes.

Domaines: Probabilités [math.PR]

S. Vakeroudis. On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity. SIAM Theory of Probability and its Applications, 2012, 56 (3), pp.485-507. ⟨10.1137/S0040585X97985546⟩. ⟨hal-00754979⟩