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

Abstract

We deal with a complex-valued Ornstein–Uhlenbeck (OU) process with parameter 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 with the skew product representation of its driving planar Brownian motion (BM) under a new deterministic time scale. We present the stochastic differential equations for the radial and for the winding process. Moreover, we obtain the large time (analogue of Spitzer's Theorem for BM in the complex plane) and the small time asymptotics for the winding and for the radial process, 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 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 J. Bertoin and W. Werner [Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. Sém. Prob., XXVIII, Lect. Notes in Mathematics 1583, Springer, Berlin Heidelberg New York, 1994, pp. 138–152] 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 process and we obtain similar results for its (well-defined) winding and radial process.