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 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.

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 The Resolvent and Expected Local Times for Markov-Modulated Brownian Motion with Phase-Dependent Termination Rates

2013 ◽  
Vol 50 (2) ◽  
pp. 430-438 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
Hazard Rate ◽  
Local Times ◽  
Insurance Risk ◽  
The Matrix ◽  
Markov Modulated

We consider a Markov-modulated Brownian motion (MMBM) with phase-dependent termination rates, i.e. while in a phase i the process terminates with a constant hazard rate ri ≥ 0. For such a process, we determine the matrix of expected local times (at zero) before termination and hence the resolvent. The results are applied to some recent questions arising in the framework of insurance risk. We further provide expressions for the resolvent and the local times at zero of an MMBM reflected at its infimum.

The Resolvent and Expected Local Times for Markov-Modulated Brownian Motion with Phase-Dependent Termination Rates

2013 ◽  
Vol 50 (02) ◽  
pp. 430-438 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
Hazard Rate ◽  
Local Times ◽  
Insurance Risk ◽  
The Matrix ◽  
Markov Modulated

We consider a Markov-modulated Brownian motion (MMBM) with phase-dependent termination rates, i.e. while in a phase i the process terminates with a constant hazard rate r i ≥ 0. For such a process, we determine the matrix of expected local times (at zero) before termination and hence the resolvent. The results are applied to some recent questions arising in the framework of insurance risk. We further provide expressions for the resolvent and the local times at zero of an MMBM reflected at its infimum.

scholarly journals Occupation Times for Markov-Modulated Brownian Motion

2012 ◽  
Vol 49 (02) ◽  
pp. 549-565 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Brownian Motion ◽  
First Passage Time ◽  
Passage Time ◽  
Laplace Transforms ◽  
Occupation Times ◽  
First Passage ◽  
Scale Functions ◽  
Positive Variation ◽  
Markov Modulated

In this paper we determine the distributions of occupation times of a Markov-modulated Brownian motion (MMBM) in separate intervals before a first passage time or an exit from an interval. We derive the distributions in terms of their Laplace transforms, and we also distinguish between occupation times in different phases. For MMBMs with strictly positive variation parameters, we further propose scale functions.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

scholarly journals The Class of Distributions Associated with the Generalized Pollaczek-Khinchine Formula

2012 ◽  
Vol 49 (3) ◽  
pp. 883-887 ◽  
Author(s):  
Offer Kella
Keyword(s):  
Brownian Motion ◽  
Stationary Distribution ◽  
Lévy Process ◽  
Random Variables ◽  
Levy Process ◽  
Independent Random Variables ◽  
Reflected Brownian Motion ◽  
Distribution Of The Maximum ◽  
Workload Distribution ◽  
Stationary Workload

The goal is to identify the class of distributions to which the distribution of the maximum of a Lévy process with no negative jumps and negative mean (equivalently, the stationary distribution of the reflected process) belongs. An explicit new distributional identity is obtained for the case where the Lévy process is an independent sum of a Brownian motion and a general subordinator (nondecreasing Lévy process) in terms of a geometrically distributed sum of independent random variables. This generalizes both the distributional form of the standard Pollaczek-Khinchine formula for the stationary workload distribution in the M/G/1 queue and the exponential stationary distribution of a reflected Brownian motion.

scholarly journals α-time fractional Brownian motion: PDE connections and local times

2012 ◽  
Vol 16 ◽  
pp. 1-24 ◽  
Author(s):  
Erkan Nane ◽  
Dongsheng Wu ◽  
Yimin Xiao
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Times
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