A Brownian motion with two reflecting barriers and Markov-modulated speed

2004 ◽  
Vol 41 (04) ◽  
pp. 1237-1242 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Linear Algebra ◽  
Stationary Distribution ◽  
The State ◽  
Long Run ◽  
Markov Modulated ◽  
Two Barriers ◽  
Reflecting Barriers

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

A Brownian motion with two reflecting barriers and Markov-modulated speed

2004 ◽  
Vol 41 (4) ◽  
pp. 1237-1242 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Linear Algebra ◽  
Stationary Distribution ◽  
The State ◽  
Long Run ◽  
Markov Modulated ◽  
Two Barriers ◽  
Reflecting Barriers

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

scholarly journals Markov-Modulated Brownian Motion with Two Reflecting Barriers

2010 ◽  
Vol 47 (04) ◽  
pp. 1034-1047 ◽  
Author(s):  
Jevgenijs Ivanovs
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Simple Form ◽  
Transient Behavior ◽  
Local Times ◽  
Probabilistic Argument ◽  
Markov Modulated ◽  
Reflecting Barriers

We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.

scholarly journals Markov-Modulated Brownian Motion with Two Reflecting Barriers

2010 ◽  
Vol 47 (4) ◽  
pp. 1034-1047 ◽  
Author(s):  
Jevgenijs Ivanovs
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Simple Form ◽  
Transient Behavior ◽  
Local Times ◽  
Probabilistic Argument ◽  
Markov Modulated ◽  
Reflecting Barriers

We consider a Markov-modulated Brownian motion reflected to stay in a strip [0, B]. The stationary distribution of this process is known to have a simple form under some assumptions. We provide a short probabilistic argument leading to this result and explain its simplicity. Moreover, this argument allows for generalizations including the distribution of the reflected process at an independent, exponentially distributed epoch. Our second contribution concerns transient behavior of the model. We identify the joint law of the processes defining the model at inverse local times.

scholarly journals Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

2021 ◽  
Vol 58 (2) ◽  
pp. 372-393
Author(s):  
H. M. Jansen
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Regime Switching ◽  
Sufficient Conditions ◽  
The State ◽  
Stochastic Integrals ◽  
Occupation Measure ◽  
Generator Matrix ◽  
Markov Modulated ◽  
Erlang Loss

AbstractOur aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Evaluating basketball player’s rotation line-ups performance via statistical markov chain modelling

2021 ◽  
pp. 174795412110090
Author(s):  
Pavlos Kolias ◽  
Nikolaos Stavropoulos ◽  
Alexandra Papadopoulou ◽  
Theodoros Kostakidis
Keyword(s):  
Markov Chain ◽  
Markov Chain Model ◽  
Specific Rotation ◽  
Multiple Regression Model ◽  
Chain Model ◽  
Know How ◽  
Basketball Game ◽  
Long Run ◽  
Rotation Line ◽  
Rating Category

Coaches in basketball often need to know how specific rotation line-ups perform in either offense or defense and choose the most efficient formation, according to their specific needs. In this research, a sample of 1131 ball possession phases of Greek Basket League was utilized, in order to estimate the offensive and defensive performance of each formation. Offensive and defensive ratings for each formation were calculated as a function of points scored or received, respectively, over possessions, where possessions were estimated using a multiple regression model. Furthermore, a Markov chain model was implemented to estimate the probabilities of the associated formation’s performance in the long run. The model could allow us to distinguish between overperforming and underperforming formations and revealed the probabilities over the evolution of the game, for each formation to be in a specific rating category. The results indicated that the most dominant formation, in terms of offense, is Point Guard-Point Guard-Small Forward-Power Forward-Center, while defensively schema Point Guard-Shooting Guard-Small Forward-Center-Center had the highest rating. Such results provide information, which could operate as a supplementary tool for the coach’s decisions, related to which rotation line-up patterns are mostly suitable during a basketball game.

Approximating the stationary distribution of an infinite stochastic matrix

1991 ◽  
Vol 28 (1) ◽  
pp. 96-103 ◽  
Author(s):  
Daniel P. Heyman
Keyword(s):  
Markov Chain ◽  
Steady State ◽  
Stationary Distribution ◽  
Numerical Approximation ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Finite System ◽  
Balance Equations ◽  
The Given ◽  
Steady State Balance

We are given a Markov chain with states 0, 1, 2, ···. We want to get a numerical approximation of the steady-state balance equations. To do this, we truncate the chain, keeping the first n states, make the resulting matrix stochastic in some convenient way, and solve the finite system. The purpose of this paper is to provide some sufficient conditions that imply that as n tends to infinity, the stationary distributions of the truncated chains converge to the stationary distribution of the given chain. Our approach is completely probabilistic, and our conditions are given in probabilistic terms. We illustrate how to verify these conditions with five examples.

scholarly journals Assessing Changes in Soil Erosion Rates: A Markov Chain Analysis

1999 ◽  
Vol 31 (3) ◽  
pp. 611-622 ◽  
Author(s):  
Rhonda Skaggs ◽  
Soumen Ghosh
Keyword(s):  
Markov Chain ◽  
Soil Erosion ◽  
Great Plains ◽  
Markov Chain Analysis ◽  
Conservation Policies ◽  
Erosion Rates ◽  
National Resources Inventory ◽  
Long Run ◽  
Chain Analysis ◽  
Soil Erosion Rates

AbstractMarkov chain analysis (one-step and long-run) is applied to the National Resources Inventory (NRI) database to evaluate changes in wind-based soil erosion rates over time. The research compares changes in soil erosion rates between NRI sample sites with and without applied conservation practices for a random sample of Great Plains counties. No significant differences between sites are found for half of the counties evaluated. The effectiveness and efficiency of conservation policies are thus questioned in light of these research results.

Institutional variations in practice of demand responsive approach: evidence from rural water supply in India

Water Policy ◽  
2014 ◽  
Vol 16 (4) ◽  
pp. 650-668 ◽  
Author(s):  
Satyapriya Rout
Keyword(s):  
Drinking Water ◽  
Water Supply ◽  
Rural Areas ◽  
The State ◽  
Indian Society ◽  
Institution Building ◽  
Rural Water Supply ◽  
Long Run ◽  
Rural Water ◽  
Village Communities

The paper reports the main findings of a study, designed to develop a better understanding of institutional variations in working with the demand responsive approach (DRA) in rural water supply in the state of Odisha in India. Data for the paper were collected from twelve village communities, where water is being supplied either through their community based institution or through the local government institution of the Gram Panchayat. The findings suggest that the two types of institutions performed differently in implementing the DRA. It depicted that the DRA under the broader rubric of institutional reforms in the water sector has failed to address the question of social inequality, and rather had reinforced and extended the already existing inequity of Indian society to access to safe and secure drinking water in rural areas. The study highlights that participation, cost recovery, full operation and management transfer may be an efficient proposition, but not sustainable in the long run without proper investment in institution building and support from the state, especially in provisioning of basic services like drinking water to rural poor.

scholarly journals Stationary distribution Markov chain for Covid-19 pandemic

2021 ◽  
Vol 1722 ◽  
pp. 012084
Author(s):  
A L H Achmad ◽  
Mahrudinda ◽  
B N Ruchjana
Keyword(s):  
Markov Chain ◽  
Stationary Distribution
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