Superpositions of Ornstein-Uhlenbeck type processes and related topics








    The lecture is based on a joint work with O.E. Barndorff ľNielsen (Aarhus University) [1].  In the fields of finance, distributions of logarithmic asset returns can often be fitted extremely well by the normal inverse Gaussian distribution or more general infinitely divisible distributions (see Barndorff-Nielsen (1998, 2001), Barndorff-Nielsen and Sheppard (2001) and references therein).

    Another issue in modelling economic time series is that their sample autocorrelation function may have non-negligible values at large lags. This phenomenon is known as long range dependence (long memory or strong dependence).

    On the other hand stochastic processes with infinitely divisible marginal distributions and long range dependence have considerable potential for stochastic modelling of observational series from a wide range of fields, such as turbulence or anomalous diffusion. These ubiquitous phenomena call for development of reasonable models which can be integrated into economic and financial theory as well as theories of turbulence or anomalous diffusion.

    This paper is motivated by the papers of Barndorff-Nielsen (1998, 2001) in which stationary processes of Ornstein-Uhlenbeck (OU) type with long-range dependence and infinitely divisible marginal distributions are constructed. These processes may, in particular, have the normal inverse Gaussian distribution as one-dimensional marginal law.

    In the present paper we discuss, within the above-mentioned framework, several new instances of continuous time strictly stationary processes whose autocorrelation functions and spectra have a simple explicit form and exhibit long-range dependence and whose marginal laws are simple and tractable. We also discussed statistical inference problems for such a processes  [2].

 We also discussed stochastic processes processes with long-range dependence and integer valued marginal distributions (such as Negative Binomial distribution) [3].


1.     Barndorff-Nielsen, O.E. and Leonenko, N.N. Spectral properties of superpositions of Ornstein-Uhlenbeck type processes, Methodology and Computing in Applied Probability, 7 (2005), 335-352 (A)

2.      Leonenko, N.N. and Taufer, E. Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion, Stochastics, 77 (2005), 477-499 (A)

3.      Leonenko, N.N., Savani, V.  and  Zhigljavsky, A. (2007) Autoregressive Negative Binomial processes, Les Annales de l`I.S.U.P. , 2007, vol. 51, Fasc. 1-2, 25-47 (A)