Non-Gaussian Ornstein–Uhlenbeck-based Models and Some of Their Uses in Financial Economics

Abstract

Summary. Non-Gaussian processes of Ornstein±Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for¯exible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of ®nance and econometrics. We construct continuous time stochastic volatility models for ®nancial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to ®nancial data and theory.

Stationarity and Ergodicity for an Affine Two-Factor Model

“…The set of affine processes contains a large class of important Markov processes such as continuous state branching processes and OrsteinUhlenbeck processes. Further, a lot of models in financial mathematics are affine such as the Heston model [18], the model of Barndorff-Nielsen and Shephard [3] or the model due to Carr and Wu [6]. A precise mathematical formulation and a complete characterization of regular affine processes are due to Duffie et al [12].…”

Martingale-Valued Measures, Ornstein-Uhlenbeck Processes with Jumps and Operator Self-Decomposability in Hilbert Space

“…We remark that, in the finite dimensional case, Ornstein-Uhlenbeck processes driven by non-Gaussian Lévy processes have recently been applied to the construction of self-similar processes via the Lamperti transform ( [18]) and to models of stochastic volatility in the theory of option pricing [5], [31]. In the latter case, it may be that the infinite dimensional model as considered here, is more appropriate, as it can approximate the very large number of incremental market activities which lead to volatility change.…”

Estimating Functions for Discretely Sampled Diffusion-Type Models

“…where R * (t; θ) = t 0 s 0 r(u; θ)duds; see Barndorff-Nielsen & Shephard (2001). For numerical calculations it is perhaps more useful that Cov θ X 2 n , X 2 n+i = ω(θ)…”