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

2010 ◽  
Vol 47 (02) ◽  
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.

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.

scholarly journals First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type

2008 ◽  
Vol 45 (03) ◽  
pp. 779-799 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Inverse Function ◽  
First Passage Times ◽  
Phase Type Distribution ◽  
First Passage ◽  
Additive Processes ◽  
Special Cases ◽  
Phase Type ◽  
Spectrally Negative Lévy Processes ◽  
Passage Times ◽  
Markov Additive Processes

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

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.

scholarly journals First Passage Times for Markov Additive Processes with Positive Jumps of Phase Type

2008 ◽  
Vol 45 (3) ◽  
pp. 779-799 ◽  
Author(s):  
Lothar Breuer
Keyword(s):  
Inverse Function ◽  
First Passage Times ◽  
Phase Type Distribution ◽  
First Passage ◽  
Additive Processes ◽  
Special Cases ◽  
Phase Type ◽  
Spectrally Negative Lévy Processes ◽  
Passage Times ◽  
Markov Additive Processes

The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

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

2010 ◽  
Vol 47 (04) ◽  
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.

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 (2) ◽  
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.

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