Ornstein–Uhlenbeck Processes and Extensions

Abstract

This paper surveys a class of Generalised Ornstein-Uhlenbeck (GOU) processes associated with Lévy processes, which has been recently much analysed in view of its applications in the financial modelling area, among others. We motivate the Lévy GOU by reviewing the framework already well understood for the "ordinary" (Gaussian) Ornstein-Uhlenbeck process, driven by Brownian motion; thus, defining it in terms of a stochastic differential equation (SDE), as the solution of this SDE, or as a time changed Brownian motion. Each of these approaches has an analogue for the GOU. Only the second approach, where the process is defined in terms of a stochastic integral, has been at all closely studied, and we take this as our definition of the GOU (see Eq. (12) below). The stationarity of the GOU, thus defined, is related to the convergence of a class of "Lévy integrals", which we also briefly review. The statistical properties of processes related to or derived from the GOU are also currently of great interest, and we mention some of the research in this area. In practise, we can only observe a discrete sample over a finite time interval, and we devote some attention to the associated issues, touching briefly on such topics as an autoregressive representation connected with a discretely sampled GOU, discrete-time perpetuities, self-decomposability, self-similarity, and the Lamperti transform. Some new statistical methodology, derived from a discrete approximation procedure, is applied to a set of financial data, to illustrate the possibilities.

Pronóstico de un título de renta fija en Colombia

“…La transformación de Girsanov para el movimiento fraccional se produce de la misma manera que para el movimiento browniano, teniendo en cuenta que se trabaja para modelos de tasa de interés y que se empleará para estimar la tasa de interés con ayuda del proceso fraccional Ornstein Uhlenbeck, Maller (2009).…”

“…Partiendo de la solución que ofrece la ecuación (1.20), al mantener la persistencia de los datos, este modelo deja de ser un proceso markoviano Maller, Müller, and Szimayer, (2009), y por lo tanto se obtiene un proceso Ornstein Uhlenbeck, con la variación que se observa en la siguiente ecuación parcial diferencial:…”

Stackelberg equilibria in a continuous-time vertical contracting model with uncertain demand and delayed information

“…It is the unique nontrivial stochastic process that is stationary, Markovian, and Gaussian (see, e.g. Maller et al (2009)). It is used in financial engineering as a model for the term structure of interest rates (see, e.g.…”

“…Vasicek (1977)), and via other variants or generalisations as a model of financial time series with applications to option pricing, portfolio optimization, and risk theory; see, e.g. Nicolato and Vernardos (2003), Barndorff-Nielsen and Shephard (2001), Maller et al (2009), and the references therein. The OU process can be thought of as a continuous-time interpolation of an autoregressive process of order 1 (AR(1) process), i.e.…”

On the ruin probability of the generalised Ornstein–Uhlenbeck process in the cramér case

“…Basic properties are given by Carmona et al [8]. A general survey of the GOU process and its applications is given by Maller et al [24]. Exact conditions for no ruin (ψ(z) = 0 for some z ≥ 0) are given by Bankovsky and Sly [3] whilst conditions for certain ruin (ψ(z) = 1 for some z ≥ 0) are examined by Bankovsky [2].…”