On Hitting Times of the Winding Processes of Planar Brownian Motion and of Ornstein--Uhlenbeck Processes via Bougerol's identity

Abstract

)>IJH=?J 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 xed u > 0:with (β t , t ≥ 0) a Brownian motion, independent of β. These identities in law for 2-dimensional processes allow to study the distributions of hitting times T θ c ≡ inf{t :and more specically of

Windings of planar processes, exponential functionals and Asian options

“…c for the BM γ associated to θ Z are given by (1.2). We are now ready to state our first result; see also [27] and [29].…”

Windings of Planar Processes and Applications to the Pricing of Asian Options

“…[2,9,15,17,22,28,32]). Here, following [26] (see also [15]), we propose an easy proof based on Proposition 2.…”

“…From e.g. formula (22) with some analytic computations, we can obtain the density function of A Z T |γ| c…”

“…Concerning the exit times from a cone for complex-valued Ornstein-Uhlenbeck processes, we heve (see also [21,22,24]):…”

“…Loosely speaking, when Williams studied windings of Brownian motion, instead of working out directly the asymptotics of the winding process θ, he studied the asymptotic behaviour of this process taken at a random time, depending on θ (for similar results but with the use of a random time independent of θ, see e.g. [22]). Next, one simply remarks that the difference between the initial winding process and the subordinated process is finite, and renormalising appropriately, this difference converges to 0.…”