Sojourn times, exit times and jitter in multivariate Markov processes

1974 ◽  
Vol 6 (04) ◽  
pp. 747-756 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Continuous Time ◽  
Exit Time ◽  
Survival Function ◽  
Transient Behavior ◽  
Reliability Theory ◽  
Exit Times ◽  
Sojourn Times ◽  
Ergodic Distribution ◽  
Stationary Markov Process ◽  
Good Set

To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.

Sojourn times, exit times and jitter in multivariate Markov processes

1974 ◽  
Vol 6 (4) ◽  
pp. 747-756 ◽  
Author(s):  
J. Keilson
Keyword(s):  
Continuous Time ◽  
Exit Time ◽  
Survival Function ◽  
Transient Behavior ◽  
Reliability Theory ◽  
Exit Times ◽  
Sojourn Times ◽  
Ergodic Distribution ◽  
Stationary Markov Process ◽  
Good Set

To treat the transient behavior of a system modeled by a stationary Markov process in continuous time, the state space is partitioned into good and bad states. The distribution of sojourn times on the good set and that of exit times from this set have a simple renewal theoretic relationship. The latter permits useful bounds on the exit time survival function obtainable from the ergodic distribution of the process. Applications to reliability theory and communication nets are given.

Bounds and Approximations for the Transient Behavior of Continuous-Time Markov Chains

1989 ◽  
Vol 3 (2) ◽  
pp. 175-198 ◽  
Author(s):  
Bok Sik Yoon ◽  
J. George Shanthikumar
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Transient Behavior ◽  
Passage Time ◽  
Lower And Upper Bounds ◽  
Simple Method ◽  
Continuous Time Markov Chains ◽  
Recent Addition ◽  
Dependent Probability

Discretization is a simple, yet powerful tool in obtaining time-dependent probability distribution of continuous-time Markov chains. One of the most commonly used approaches is uniformization. A recent addition to such approaches is an external uniformization technique. In this paper, we briefly review these different approaches, propose some new approaches, and discuss their performances based on theoretical bounds and empirical computational results. A simple method to get lower and upper bounds for first passage time distribution is also proposed.

The Relaxation time for truncated birth-death processes

1987 ◽  
Vol 1 (4) ◽  
pp. 367-381 ◽  
Author(s):  
Julian Keilson ◽  
Ravi Ramaswamy
Keyword(s):  
Relaxation Time ◽  
Exit Time ◽  
Relaxation Times ◽  
Dual Process ◽  
Death Process ◽  
Exit Times ◽  
Initial State ◽  
Death Rates ◽  
Absorbing States ◽  
Birth Death

The relaxation time for an ergodic Markov process is a measure of the time until ergodicity is reached from its initial state. In this paper the relaxation time for an ergodic truncated birth-death process is studied. It is shown that the relaxation time for such a process on states {0,1, …, N} is the quasi-stationary exit time from the set {,2, …, N{0,1,…, N, N + 1} with two-sided absorption at states 0 and N + 1. The existence of such a dual process has been observed by Siegmund [15] for stochastically monotone Markov processes on the real line. Exit times for birth- death processes with two absorbing states are studied and an efficient algorithm for the numerical evaluation of mean exit times is presented. Simple analytical lower bounds for the relaxation times are obtained. These bounds are numerically accessible. Finally, the sensitivity of the relaxation time to variations in birth and death rates is studied.

Monotone and associated Markov chains, with applications to reliability theory

1987 ◽  
Vol 24 (03) ◽  
pp. 679-695 ◽  
Author(s):  
Bo Henry Lindqvist
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Reliability Theory ◽  
State Spaces ◽  
Partially Ordered ◽  
Ordered State

We study monotone and associated Markov chains on finite partially ordered state spaces. Both discrete and continuous time, and both time-homogeneous and time-inhomogeneous chains are considered. The results are applied to binary and multistate reliability theory.

scholarly journals Exact simulation of first exit times for one-dimensional diffusion processes

2020 ◽  
Vol 54 (3) ◽  
pp. 811-844
Author(s):  
Samuel Herrmann ◽  
Cristina Zucca
Keyword(s):  
Diffusion Processes ◽  
Exit Time ◽  
Convergent Series ◽  
Mathematical Finance ◽  
Exit Times ◽  
Rejection Sampling ◽  
Girsanov Transformation ◽  
One Dimensional ◽  
Usual Procedure ◽  
First Exit Times

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.

scholarly journals Tails of exit times from unstable equilibria on the line

2020 ◽  
Vol 57 (2) ◽  
pp. 477-496
Author(s):  
Yuri Bakhtin ◽  
Zsolt Pajor-Gyulai
Keyword(s):  
Vector Field ◽  
Malliavin Calculus ◽  
Elementary Proof ◽  
Exit Time ◽  
Exit Times ◽  
Smooth Vector ◽  
Small Noise ◽  
One Dimensional ◽  
Smooth Vector Field ◽  
Exit Distribution

AbstractFor a one-dimensional smooth vector field in a neighborhood of an unstable equilibrium, we consider the associated dynamics perturbed by small noise. We give a revealing elementary proof of a result proved earlier using heavy machinery from Malliavin calculus. In particular, we obtain precise vanishing noise asymptotics for the tail of the exit time and for the exit distribution conditioned on atypically long exits. We also discuss our program on rare transitions in noisy heteroclinic networks.

scholarly journals On first exit times and their means for Brownian bridges

2019 ◽  
Vol 56 (3) ◽  
pp. 701-722 ◽  
Author(s):  
Christel Geiss ◽  
Antti Luoto ◽  
Paavo Salminen
Keyword(s):  
Brownian Bridge ◽  
Exit Time ◽  
Three Dimensional ◽  
Exit Times ◽  
First Exit Time ◽  
Binomial Tree ◽  
Brownian Bridges ◽  
Bessel Bridge ◽  
The Mean ◽  
First Exit Times

AbstractFor a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.

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