On the windings of complex-valued Ornstein-Uhlenbeck processes driven by a Brownian motion and by a Stable process

TL;DR

This paper investigates the winding behavior of complex-valued Ornstein-Uhlenbeck processes driven by Brownian motion and Stable processes, deriving SDEs, asymptotics, and limit theorems for windings and radial components.

Contribution

It introduces a skew product representation for OU processes, analyzes windings via SDEs, and extends results to Stable process-driven OU processes, including asymptotics and limit theorems.

Findings

Large time asymptotics for windings analogous to Spitzer's theorem

Decomposition of windings into 'small' and 'big' windings with only 'small' contributing asymptotically

Extension of winding analysis to OU processes driven by Stable processes

Abstract

We deal with a complex-valued Ornstein-Uhlenbeck (OU) process with parameter $\lambda\in\mathbb{R}$starting from a point different from 0 and the way that it winds around the origin.The starting point of this paper is the skew product representation for an OU process which is associated to the skew product representationof its driving planar Brownian motion under a new deterministic time scale.We present the stochastic differential equations (SDEs)for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for Brownian motion in the complex plane) and the small time asymptotics for the winding and for the radialprocess, and we explore the exit time from a cone for a 2-dimensional OU process.Some Limit Theorems concerning the angle of the cone (when our process winds in a cone) and the parameter $\lambda$ are also presented.Furthermore, we discuss the decomposition of the winding process of a complex-valued OU process in "small" and "big" windings,where, for the "big" windings, we use some results already obtained by Bertoin and Werner in \cite{BeW94},and we show that only the "small" windings contribute in the large time limit.Finally, we study the windings of a complex-valued OU process driven by a Stable processand we obtain similar results for its (well-defined) winding and radial process.