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Discrete Ornstein-Uhlenbeck process in a stationary dynamic enviroment

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The thesis is devoted to the study of solutions to the following linear recursion:

\beq

X_{n+1}=\gamma X_n+ \xi_n,

\feq

where $\gamma \in (0,1)$ is a constant and $(\xi_n)_{n\in\zz}$

is a stationary and ergodic sequence of normal variables with \emph{random} means and

variances. More precisely, we assume that

\beq

\xi_n=\mu_n+\sigma_n\veps_n,

\feq

where $(\veps)_{n\in\zz}$ is an i.i.d. sequence of standard normal variables

and $(\mu_n,\sigma_n)_{n\in\zz}$ is a stationary and ergodic process

independent of $(\veps_n)_{n\in\zz},$ 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.

\par

The solution to the above recursion is well understood in the classical case, when $\xi_n$ 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.

\par

In this thesis we mostly focus on the study of general features, common for all solutions to the recursion

with the innovation/error term $\xi_n$ modulated as above by a random environment $(\mu_n,\sigma_n),$

regardless the distribution of the environment. In particular, we study asymptotic behavior of the solution

when $\gamma$ approaches $1.$ In addition, we investigate the asymptotic behavior of the extreme values

$M_n=\max_{1\leq k\leq n} X_k$ and the partial sums $S_n=\sum_{k=1}^n X_k.$ The case of Markov-dependent

environments will be studied in more detail elsewhere.

\par

The existence of general patterns in the long-term behavior of $X_n,$ 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.