Discrete Ornstein-Uhlenbeck process in a stationary dynamic enviroment

The thesis is devoted to the study of solutions to the following linear recursion: Xn+1 = γXn + ξn, where γ ∈ (0, 1) is a constant and (ξn)n∈Z is a stationary and ergodic sequence of normal variables with random means and variances. More precisely, we assume that ξn = μn + σnεn, where (ε)n∈Z is an i.i.d. sequence of standard normal variables and (μn, σn)n∈Z is a stationary and ergodic process independent of (εn)n∈Z, which serves as an exogenous dynamic environment for the model. This is an example of a so called SV (stands for stochastic variance or stochastic volatility) time-series model. We refer to the stationary solution of this recursion as a discrete Ornstein-Uhlenbeck process in a stationary dynamic environment. The solution to the above recursion is well understood in the classical case, when ξn form an i.i.d. sequence. When the pairs mean and variance form a two-component finite-state Markov process, the recursion can be thought as a discrete-time analogue of the Langevin equation with regime switches, a continuous-time model of a type which is widely used in econometrics to analyze financial time series. In this thesis we mostly focus on the study of general features, common for all solutions to the recursion with the innovation/error term ξn modulated as above by a random environment (μn, σn), regardless the distribution of the environment. In particular, we study asymptotic behavior of the solution when γ approaches 1. In addition, we investigate the asymptotic behavior of the extreme values Mn = max1≤k≤nXk and the partial sums Sn = ∑n k=1Xk. The case of Markov-dependent environments will be studied in more detail elsewhere. The existence of general patterns in the long-term behavior of Xn, independent of a particular choice of the environment, is a manifestation of the universality of the underlying mathematical framework. It turns out that the setup allows for a great flexibility in modeling yet maintaining tractability, even when is considered in its full generality. We thus believe that the model is of interest from both theoretical as well as practical points of views; in particular, for modeling financial time series.