Tail Asymptotics of the Occupation Measure for a Markov Additive Process with anM/G/1-Type Background Process

2010 ◽  
Vol 26 (3) ◽  
pp. 463-486 ◽  
Author(s):  
Masahiro Kobayashi ◽  
Masakiyo Miyazawa ◽  
Yiqiang Q. Zhao
Keyword(s):  
Tail Asymptotics ◽  
Occupation Measure ◽  
Background Process ◽  
Additive Process ◽  
Markov Additive Process

scholarly journals TRAFFIC GENERATED BY A SEMI-MARKOV ADDITIVE PROCESS

2010 ◽  
Vol 25 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Joke Blom ◽  
Michel Mandjes
Keyword(s):  
Positive Solution ◽  
Linear Equations ◽  
Levy Processes ◽  
System Of Linear Equations ◽  
Unique Positive Solution ◽  
Exponential Distributions ◽  
Background Process ◽  
Additive Process ◽  
Markov Additive Process ◽  
Sojourn Times

We consider a semi-Markov additive process A(·)—that is, a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the Lévy processes Xi(·), which describe the evolution of A(·) while the background process is in state i, be increasing, it is shown how double transforms of the type $\vint_{0}^{\infty} e^{-qt}\, {\open E} \lsqb e^{-\alpha A(t)}\, {d}t \rsqb$ can be computed. It turns out that these follow, for given nonnegative α and q, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well.

scholarly journals On the finiteness and tails of perpetuities under a Lamperti–Kiu MAP

2021 ◽  
Vol 58 (4) ◽  
pp. 1086-1113
Author(s):  
Larbi Alili ◽  
David Woodford
Keyword(s):  
Markov Chain ◽  
Tail Asymptotics ◽  
Additive Process ◽  
Markov Additive Process ◽  
Chain Component ◽  
Exponential Functional ◽  
Cramér’S Condition

AbstractConsider a Lamperti–Kiu Markov additive process $(J, \xi)$ on $\{+, -\}\times\mathbb R\cup \{-\infty\}$, where J is the modulating Markov chain component. First we study the finiteness of the exponential functional and then consider its moments and tail asymptotics under Cramér’s condition. In the strong subexponential case we determine the subexponential tails of the exponential functional under some further assumptions.

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 $\omega(\cdot,\cdot)$ .

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