scholarly journals First Passage Time of a Markov Chain That Converges to Bessel Process

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Moussa Kounta
Keyword(s):  
Markov Chain ◽  
First Passage Time ◽  
Passage Time ◽  
Bessel Process ◽  
Hitting Time ◽  
The Other ◽  
First Hitting Time ◽  
First Passage ◽  
Discrete Markov Chain

We investigate the probability of the first hitting time of some discrete Markov chain that converges weakly to the Bessel process. Both the probability that the chain will hit a given boundary before the other and the average number of transitions are computed explicitly. Furthermore, we show that the quantities that we obtained tend (with the Euclidian metric) to the corresponding ones for the Bessel process.

Dynamical behavior of simplified FitzHugh–Nagumo neural system with time delay driven by Lévy noise

2021 ◽  
pp. 2150240
Author(s):  
Weida Qiu ◽  
Yongfeng Guo ◽  
Xiuxian Yu
Keyword(s):  
Time Delay ◽  
First Passage Time ◽  
Dynamical Behavior ◽  
Passage Time ◽  
Neural System ◽  
The Other ◽  
Lévy Noise ◽  
First Passage ◽  
Levy Noise ◽  
System With Time Delay

In this paper, the dynamical behavior of the FitzHugh–Nagumo (FHN) neural system with time delay driven by Lévy noise is studied from two aspects: the mean first-passage time (MFPT) and the probability density function (PDF) of the first-passage time (FPT). Using the Janicki–Weron algorithm to generate the Lévy noise, and through the order-4 Runge–Kutta algorithm to simulate the FHN system response, the time that the system needs from one stable state to the other one is tracked in the process. Using the MATLAB software to simulate the process above 20,000 times and recording the PFTs, the PDF of the FPT and the MFPT is obtained. Finally, the effects of the Lévy noise and time-delay on the FPT are discussed. It is found that the increase of both time-delay feedback intensity and Lévy noise intensity can promote the transition of the particle from the resting state to the excited state. However, the two parameters produce the opposite effects in the other direction.

scholarly journals Number of successes in Markov trials

1988 ◽  
Vol 20 (3) ◽  
pp. 677-680 ◽  
Author(s):  
U. Narayan Bhat ◽  
Ram Lal
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
First Passage Time ◽  
Passage Time ◽  
Chain Model ◽  
First Passage

Markov trials are a sequence of dependent trials with two outcomes, success and failure, which are the states of a Markov chain. The distribution of the number of successes in n Markov trials and the first-passage time for a specified number of successes are obtained using an augmented Markov chain model.

scholarly journals Deviation bounds for the first passage time in the frog model

2019 ◽  
Vol 51 (01) ◽  
pp. 184-208 ◽  
Author(s):  
Naoki Kubota
Keyword(s):  
Random Walks ◽  
Large Deviation ◽  
First Passage Time ◽  
Passage Time ◽  
The Other ◽  
Active Particle ◽  
First Passage ◽  
Cubic Lattice ◽  
Concentration Bounds ◽  
Frog Model

AbstractWe consider the so-called frog model with random initial configurations. The dynamics of this model are described as follows. Some particles are randomly assigned to any site of the multidimensional cubic lattice. Initially, only particles at the origin are active and these independently perform simple random walks. The other particles are sleeping and do not move at first. When sleeping particles are hit by an active particle, they become active and start moving in a similar fashion. The aim of this paper is to derive large deviation and concentration bounds for the first passage time at which an active particle reaches a target site.

scholarly journals On the level crossing of multi-dimensional delayed renewal processes

1997 ◽  
Vol 10 (4) ◽  
pp. 355-361 ◽  
Author(s):  
Jewgeni H. Dshalalow
Keyword(s):  
Renewal Process ◽  
Stochastic Models ◽  
First Passage Time ◽  
Passage Time ◽  
The Other ◽  
Renewal Processes ◽  
Active Components ◽  
Level Crossing ◽  
First Passage ◽  
Informal Discussion

The paper studies the behavior of an (l+3)th-dimensional, delayed renewal process with dependent components, the first three (called active) of which are to cross one of their respective thresholds. More specifically, the crossing takes place when at least one of the active components reaches or exceeds its assigned level. The values of the other two active components, as well as the rest of the components (passive), are to be registered. The analysis yields the joint functional of the “crossing level” and other characteristics (some of which can be interpreted as the first passage time) in a closed form, refining earlier results of the author. A brief, informal discussion of various applications to stochastic models is presented.

scholarly journals Number of successes in Markov trials

1988 ◽  
Vol 20 (03) ◽  
pp. 677-680 ◽  
Author(s):  
U. Narayan Bhat ◽  
Ram Lal
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
First Passage Time ◽  
Passage Time ◽  
Chain Model ◽  
First Passage

Markov trials are a sequence of dependent trials with two outcomes, success and failure, which are the states of a Markov chain. The distribution of the number of successes in n Markov trials and the first-passage time for a specified number of successes are obtained using an augmented Markov chain model.

On passage and conditional passage times for Markov chains in continuous time

1988 ◽  
Vol 25 (02) ◽  
pp. 279-290 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
First Passage ◽  
Reversible Markov Chain ◽  
Irreducible Markov Chain ◽  
Passage Times

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let kTij ( jTik ) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij ( jmik ) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik , and is also discussed, where , and are conditional passage times of the reversed process of X(t).

On passage and conditional passage times for Markov chains in continuous time

1988 ◽  
Vol 25 (2) ◽  
pp. 279-290 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Death Process ◽  
First Passage ◽  
Reversible Markov Chain ◽  
Irreducible Markov Chain ◽  
Passage Times

Let X(t) be a temporally homogeneous irreducible Markov chain in continuous time defined on . For k < i < j, let H = {k + 1, ···, j − 1} and let kTij (jTik) be the upward (downward) conditional first-passage time of X(t) from i to j(k) given no visit to . These conditional passage times are studied through first-passage times of a modified chain HX(t) constructed by making the set of states absorbing. It will be shown that the densities of kTij and jTik for any birth-death process are unimodal and the modes kmij (jmik) of the unimodal densities are non-increasing (non-decreasing) with respect to i. Some distribution properties of kTij and jTik for a time-reversible Markov chain are presented. Symmetry among kTij, jTik, and is also discussed, where , and are conditional passage times of the reversed process of X(t).

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