First Hitting Place Probabilities for a Discrete Version of the Ornstein-Uhlenbeck Process

A Markov chain with state space {0,…,N} and transition probabilities depending on the current state is studied. The chain can be considered as a discrete Ornstein-Uhlenbeck process. The probability that the process hits N before 0 is computed explicitly. Similarly, the probability that the process hits N before −M is computed in the case when the state space is {−M,…,0,…,N} and the transition probabilities pi,i+1 are not necessarily the same when i is positive and i is negative

First hitting problems for Markov chains that converge to a geometric Brownian motion

We consider a discrete-time Markov chain with state space {1,1+∆x,...,1+k∆x = N}. We compute explicitly the probability pj that the chain, starting from 1 + j∆x, will hit N before 1, as well as the expected number dj of transitions needed to end the game. In the limit when ∆x and the time ∆t between the transitions decrease to zero appropriately, the Markov chain tends to a geometric Brownian motion. We show that pj and dj∆t tend to the corresponding quantities for the geometric Brownian motion

Discrete Ornstein-Uhlenbeck process in a stationary dynamic enviroment

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