scholarly journals Hitting probabilities in a Markov additive process with linear movements and upward jumps: Applications to risk and queueing processes

2004 ◽  
Vol 14 (2) ◽  
pp. 1029-1054 ◽  
Author(s):  
Masakiyo Miyazawa
Keyword(s):  
Additive Process ◽  
Markov Additive Process ◽  
Hitting Probabilities ◽  
Queueing Processes

scholarly journals Exact boundaries in sequential testing for phase-type distributions

2014 ◽  
Vol 51 (A) ◽  
pp. 347-358
Author(s):  
Hansjörg Albrecher ◽  
Peiman Asadi ◽  
Jevgenijs Ivanovs
Keyword(s):  
Ratio Test ◽  
Type I ◽  
Phase Type Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
Scale Functions ◽  
Phase Type ◽  
Sequential Probability ◽  
Decision Boundaries ◽  
Type Ii Errors

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

scholarly journals Exact boundaries in sequential testing for phase-type distributions

2014 ◽  
Vol 51 (A) ◽  
pp. 347-358
Author(s):  
Hansjörg Albrecher ◽  
Peiman Asadi ◽  
Jevgenijs Ivanovs
Keyword(s):  
Ratio Test ◽  
Type I ◽  
Phase Type Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
Scale Functions ◽  
Phase Type ◽  
Sequential Probability ◽  
Decision Boundaries ◽  
Type Ii Errors

Consider Wald's sequential probability ratio test for deciding whether a sequence of independent and identically distributed observations comes from a specified phase-type distribution or from an exponentially tilted alternative distribution. Exact decision boundaries for given type-I and type-II errors are derived by establishing a link with ruin theory. Information on the mean sample size of the test can be retrieved as well. The approach relies on the use of matrix-valued scale functions associated with a certain one-sided Markov additive process. By suitable transformations, the results also apply to other types of distributions, including some distributions with regularly varying tails.

scholarly journals Lundberg-type Bounds for the Joint Distribution of Surplus Immediately Before and at Ruin under a Markov-modulated Risk Model

2005 ◽  
Vol 35 (02) ◽  
pp. 351-361 ◽  
Author(s):  
Andrew C.Y. Ng ◽  
Hailiang Yang
Keyword(s):  
Regime Switching ◽  
Joint Distribution ◽  
Risk Model ◽  
Insurance Risk ◽  
Additive Process ◽  
Deficit At Ruin ◽  
Markov Additive Process ◽  
The Deficit At Ruin ◽  
Markov Modulated ◽  
Markovian Regime Switching

In this paper, we consider a Markov-modulated risk model (also called Markovian regime switching insurance risk model). Follow Asmussen (2000, 2003), by using the theory of Markov additive process, an exponential martingale is constructed and Lundberg-type upper bounds for the joint distribution of surplus immediately before and at ruin are obtained. As a natural corollary, bounds for the distribution of the deficit at ruin are obtained. We also present some numerical results to illustrate the tightness of the bound obtained in this paper.

scholarly journals First Passage of a Markov Additive Process and Generalized Jordan Chains

2010 ◽  
Vol 47 (4) ◽  
pp. 1048-1057 ◽  
Author(s):  
Bernardo D‘Auria ◽  
Jevgenijs Ivanovs ◽  
Offer Kella ◽  
Michel Mandjes
Keyword(s):  
Matrix Function ◽  
Crucial Role ◽  
Fluctuation Theory ◽  
Matrix Functions ◽  
First Passage ◽  
Additive Process ◽  
Process Map ◽  
Markov Additive Process ◽  
Analytic Matrix ◽  
The Law

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. This result provides us with a technique that can be used to derive various further identities.

Fluctuation identities for Omega-killed spectrally negative Markov additive processes and dividend problem

2020 ◽  
Vol 52 (2) ◽  
pp. 404-432
Author(s):  
Irmina Czarna ◽  
Adam Kaszubowski ◽  
Shu Li ◽  
Zbigniew Palmowski
Keyword(s):  
Insurance Risk ◽  
Additive Process ◽  
Markov Additive Process ◽  
Surplus Process ◽  
Current State ◽  
Markov Modulated ◽  
Exit Problems ◽  
State Of The Environment ◽  
Risk Processes ◽  
Omega Model

AbstractIn this paper, we solve exit problems for a one-sided Markov additive process (MAP) which is exponentially killed with a bivariate killing intensity $\omega(\cdot,\cdot)$ dependent on the present level of the process and the current state of the environment. Moreover, we analyze the respective resolvents. All identities are expressed in terms of new generalizations of classical scale matrices for MAPs. We also remark on a number of applications of the obtained identities to (controlled) insurance risk processes. In particular, we show that our results can be applied to the Omega model, where bankruptcy takes place at rate $\omega(\cdot,\cdot)$ when the surplus process becomes negative. Finally, we consider Markov-modulated Brownian motion (MMBM) as a special case and present analytical and numerical results for a particular choice of piecewise intensity function $\omega(\cdot,\cdot)$ .

scholarly journals A sample path approach to mean busy periods for Markov-modulated queues and fluids

1994 ◽  
Vol 26 (4) ◽  
pp. 1117-1121 ◽  
Author(s):  
Søren Asmussen ◽  
Mogens Bladt
Keyword(s):  
Time Reversal ◽  
Busy Period ◽  
Fluid Model ◽  
Sample Path ◽  
Steady State Distribution ◽  
State Distribution ◽  
Additive Process ◽  
Markov Additive Process ◽  
The Mean ◽  
Markov Modulated

The mean busy period of a Markov-modulated queue or fluid model is computed by an extension of the time-reversal argument connecting the steady-state distribution and the maximum of a related Markov additive process.

scholarly journals A quintuple law for Markov additive processes with phase-type jumps

2010 ◽  
Vol 47 (2) ◽  
pp. 441-458 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Joint Distribution ◽  
First Passage Times ◽  
Insurance Risk ◽  
Maximal Level ◽  
First Passage ◽  
Additive Process ◽  
Process Map ◽  
Markov Additive Process ◽  
Phase Type ◽  
Passage Times

We consider a Markov additive process (MAP) with phase-type jumps, starting at 0. Given a positive level u, we determine the joint distribution of the undershoot and overshoot of the first jump over the level u, the maximal level before this jump, the time of attaining this maximum, and the time between the maximum and the jump. The analysis is based on first passage times and time reversion of MAPs. A marginal of the derived distribution is the Gerber-Shiu function, which is of interest to insurance risk. Several examples serve to compare the present result with the literature.

Markov Additive Processes and Repeats in Sequences

2007 ◽  
Vol 44 (02) ◽  
pp. 514-527
Author(s):  
John L. Spouge
Keyword(s):  
Time Lag ◽  
Fixed Time ◽  
Biological Sequences ◽  
Additive Process ◽  
Usual Model ◽  
Markov Additive Process ◽  
Fixed Distribution ◽  
Two Parameters ◽  
Simple Repeats ◽  
Model Sequence

Computer analysis of biological sequences often detects deviations from a random model. In the usual model, sequence letters are chosen independently, according to some fixed distribution over the relevant alphabet. Real biological sequences often contain simple repeats, however, which can be broadly characterized as multiple contiguous copies (usually inexact) of a specific word. This paper quantifies inexact simple repeats as local sums in a Markov additive process (MAP). The maximum of the local sums has an asymptotic distribution with two parameters (λ and k), which are given by general MAP formulas. The general MAP formulas are usually computationally intractable, but an essential simplification in the case of repeats permits λ and k to be computed from matrices whose dimension equals the size of the relevant alphabet. The simplification applies to some MAPs where the summand distributions do not depend on consecutive pairs of Markov states as usual, but on pairs with a fixed time-lag larger than one.

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