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.

Markov-modulated linear fluid networks with Markov additive input

2002 ◽  
Vol 39 (02) ◽  
pp. 413-420 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Markov Chain ◽  
External Input ◽  
Release Rates ◽  
Constant Multiple ◽  
Additive Process ◽  
Markov Additive Process ◽  
Fluid Networks ◽  
Markov Modulated

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

Markov-modulated linear fluid networks with Markov additive input

2002 ◽  
Vol 39 (2) ◽  
pp. 413-420 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Markov Chain ◽  
External Input ◽  
Release Rates ◽  
Constant Multiple ◽  
Additive Process ◽  
Markov Additive Process ◽  
Fluid Networks ◽  
Markov Modulated

We consider a network of dams to which the external input is a multivariate Markov additive process. For each state of the Markov chain modulating the Markov additive process, the release rates are linear (constant multiple of the content level). Each unit of material processed by a given station is then divided into fixed proportions each of which is routed to another station or leaves the system. For each state of the modulating process, this routeing is determined by some substochastic matrix. We identify simple conditions for stability and show how to compute transient and stationary characteristics of such networks.

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.

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