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In this paper we develop the theory of the so-called $mathbf{W}$ and $mathbf{Z}$ scale matrices for (upwards skip-free) discrete-time and discrete-space Markov additive processes, along the lines of the analogous theory for Markov additive processes in continuous-time. In particular, we provide their probabilistic construction, identify the form of the generating function of $mathbf{W}$ and its connection with the occupation mass formula, which provides the tools for deriving semi-explicit expressions for corresponding exit problems for the upward-skip free process and its reflections, in terms the scale matrices.
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Let $mathbb{hat{E}}$ be the upper expectation of a weakly compact but non-dominated family $mathcal{P}$ of probability measures. Assume that $Y$ is a $d$-dimensional $mathcal{P}$-semimartingale under $mathbb{hat{E}}$. Given an open set $Qsubsetmathbb{R}^{d}$, the exit time of $Y$ from $Q$ is defined by [ {tau}_{Q}:=inf{tgeq0:Y_{t}in Q^{c}}. ] The main objective of this paper is to study the quasi-continuity properties of ${tau}_{Q}$ under the nonlinear expectation $mathbb{hat{E}}$. Under some additional assumptions on the growth and regularity of $Y$, we prove that ${tau}_{Q}wedge t$ is quasi-continuous if $Q$ satisfies the exterior ball condition. We also give the characterization of quasi-continuous processes and related properties on stopped processes. In particular, we get the quasi-continuity of exit times for multi-dimensional $G$-martingales, which nontrivially generalizes the previous one-dimensional result of Song.
In this article we determine the Laplace transforms of the main boundary functionals of the oscillating compound Poisson process. These are the first passage time of the level, the joint distribution of the first exit time from the interval and the value of the overshoot through the boundary. Under certain conditions we establish the asymptotic behaviour of the mentioned functionals.
We consider random walk on dynamical percolation on the discrete torus $mathbb{Z}_n^d$. In previous work, mixing times of this process for $p<p_c(mathbb{Z}^d)$ were obtained in the annealed setting where one averages over the dynamical percolation environment. Here we study exit times in the quenched setting, where we condition on a typical dynamical percolation environment. We obtain an upper bound for all $p$ which for $p<p_c$ matches the known lower bound.
We consider a sequence of additive functionals {phi_n}, set on a sequence of Markov chains {X_n} that weakly converges to a Markov process X. We give sufficient condition for such a sequence to converge in distribution, formulated in terms of the characteristics of the additive functionals, and related to the Dynkins theorem on the convergence of W-functionals. As an application of the main theorem, the general sufficient condition for convergence of additive functionals in terms of transition probabilities of the chains X_n is proved.
We analyse an additive-increase and multiplicative-decrease (aka growth-collapse) process that grows linearly in time and that experiences downward jumps at Poisson epochs that are (deterministically) proportional to its present position. This process is used for example in modelling of Transmission Control Protocol (TCP) and can be viewed as a particular example of the so-called shot noise model, a basic tool in modeling earthquakes, avalanches and neuron firings. For this process, and also for its reflect
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