Queues and Risk Processes with Dependencies

Badila, E.S.; Boxma, Onno; Resing, Jacques

doi:10.1080/15326349.2014.930603

Cited by 19 publications

(19 citation statements)

References 27 publications

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“…It has found a number of interesting applications across various fields. The advantage of working under phase-type distribution is that it allows some analytically tractable results in applications: e.g., in actuarial science (Rolski et al 1998, Albrecher and Asmussen 2010, Lin and Liu 2007, Lee and Lin 2010, option pricing (Asmussen et al 2004, Rolski et al 1998, queueing theory (Badila et al 2014, Chakravarthy and Neuts 2014, Buchholz et al 2014, Breuer and Baum 2005, Asmussen 2003, reliability theory (Assaf andLevikson 1982, Okamura andDohi 2015), and in survival analysis (Aalen 1995, Aalen andGjessing 2001). When jumps distribution of a compound Poisson process is modeled by phase-type distribution, it results in a dense class of Lévy processes, Asmussen et al (2004).…”

Section: Introductionmentioning

confidence: 99%

Distributional Properties of the Mixture of Continuous-Time Absorbing Markov Chains Moving at Different Speeds

Surya

2018

Stochastic Systems

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Abstract. This paper analyses the mixture of two continuous-time Markov chains with absorption on the same state space moving at different speeds, where the mixture occurs at a random time. Variety of associated distributional properties of the Markov mixture process are discussed, for example the transition matrix, the distribution of its lifetime and the corresponding forward intensity. Identities are given explicitly in terms of the Bayesian updates of switching probability and the intensity matrices of the underlying Markov chains despite the fact that the mixture process is not Markovian. They form nonstationary functions of time and duration and have two appealing features: the ability to capture heterogeneity and path dependence when conditioning on the available information (either full or partial) about the past history of the process until current time. In particular, the unconditional lifetime distribution forms a generalized mixture of phase-type distributions. The distribution has dense and closure properties under finite convex mixtures and finite convolutions. When the underlying Markov chains move at the same speed, in which case the mixture process reduces to a simple Markov chain, the heterogeneity and path dependence are removed, and the lifetime has the usual phase-type distribution, Neuts [Neuts MF (1975) Probability distributions of phase-type.

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Section: Introductionmentioning

confidence: 99%

Distributional Properties of the Mixture of Continuous-Time Absorbing Markov Chains Moving at Different Speeds

Surya

2018

Stochastic Systems

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“…The approach we take in this paper generalizes ideas in [5] and [6], and will allow us to extend those two studies. In Section 3 we derive a similar functional equation as in [6] for the survival function related to a 2-dimensional reserve process, but unlike [6] we do not assume that the claim intervals are exponentially distributed.…”

Section: Introductionmentioning

confidence: 95%

“…Such a correlation is quite natural; e.g., a claim event that generates very large claims could be subjected to additional administrative/regulatory delays. The type of correlation between the inter-arrival time and the vector of claim sizes is an extension to two dimensions of the dependence structure studied in [5] for a generalized Sparre-Andersen model. It involves making a rationality assumption regarding the trivariate LST of inter-arrival time and claim size vector (Assumption 2.1), which extends the case where the vector with the aforementioned components has a multivariate phase type distribution (MPH).…”

Section: Introductionmentioning

confidence: 99%

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

Badila¹,

Boxma²,

Resing³

2015

Insurance: Mathematics and Economics

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a b s t r a c tWe investigate an insurance risk model that consists of two reserves which receive income at fixed rates. Claims are being requested at random epochs from each reserve and the interclaim times are generally distributed. The two reserves are coupled in the sense that at a claim arrival epoch, claims are being requested from both reserves and the amounts requested are correlated. In addition, the claim amounts are correlated with the time elapsed since the previous claim arrival.We focus on the probability that this bivariate reserve process survives indefinitely. The infinitehorizon survival problem is shown to be related to the problem of determining the equilibrium distribution of a random walk with vector-valued increments with 'reflecting' boundary. This reflected random walk is actually the waiting time process in a queueing system dual to the bivariate ruin process.Under assumptions on the arrival process and the claim amounts, and using Wiener-Hopf factorization with one parameter, we explicitly determine the Laplace-Stieltjes transform of the survival function, c.q., the two-dimensional equilibrium waiting time distribution.Finally, the bivariate transforms are evaluated for some examples, including for proportional reinsurance, and the bivariate ruin functions are numerically calculated using an efficient inversion scheme.

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“…in actuarial science (Rolski et al [31], Albrecher and Asmussen [7], Lin and Liu [25], Lee and Lin [24]), option pricing (Asmussen et al [5], Rolski et. al [31]), queueing theory (Badila et al [9], Chakravarthy and Neuts [13], Buchholz et al [12], Breuer and Baum [11], Asmussen [6]), reliability theory (Assaf and Levikson [8], Okamura and Dohi [30]), and in survival analysis (Aalen [3], Aalen and Gjessing [2]).…”

Section: Introductionmentioning

confidence: 99%

Generalized Phase-Type Distribution and Competing Risks for Markov Mixtures Process

Surya

2016

SSRN Journal

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Phase-type distribution has been an important probabilistic tool in the analysis of complex stochastic system evolution. It was introduced by Neuts [28] in 1975. The model describes the lifetime distribution of a finite-state absorbing Markov chains, and has found many applications in wide range of areas. It was brought to survival analysis by Aalen [3] in 1995. However, the model has lacks of ability in modeling heterogeneity and inclusion of past information which is due to the Markov property of the underlying process that forms the model. We attempt to generalize the distribution by replacing the underlying by Markov mixtures process. Markov mixtures process was used to model jobs mobility by Blumen [10] et al. in 1955. It was known as the mover-stayer model describing low-productivity workers tendency to move out of their jobs by a Markov chains, while those with high-productivity tend to stay in the job. Frydman [20] later extended the model to a mixtures of finite-state Markov chains moving at different speeds on the same state space. In general the mixtures process does not have Markov property. We revisit the mixtures model [20] for mixtures of multi absorbing states Markov chains, and propose generalization of the phase-type distribution under competing risks. The new distribution has two main appealing features: it has the ability to model heterogeneity and to include past information of the underlying process, and it comes in a closed form. Built upon the new distribution, we propose conditional forward intensity which can be used to determine rate of occurrence of future events (caused by certain type) based on available * Email address: budhi.surya@vuw.ac.nz; Postal address: School of Mathematics and Statistics, Victoria University of Wellington, Gate 6 Kelburn PDE, Wellington 6140, New Zealand. arXiv:1611.03832v1 [stat.ME] 11 Nov 2016 2 B.A. Surya information. Numerical study suggests that the new distribution and its forward intensity offer significant improvements over the existing model. MSC2010 subject classifications: 60J20, 60J27, 60J28, 62N99

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Queues and Risk Processes with Dependencies

Cited by 19 publications

References 27 publications

Distributional Properties of the Mixture of Continuous-Time Absorbing Markov Chains Moving at Different Speeds

Distributional Properties of the Mixture of Continuous-Time Absorbing Markov Chains Moving at Different Speeds

Two parallel insurance lines with simultaneous arrivals and risks correlated with inter-arrival times

Generalized Phase-Type Distribution and Competing Risks for Markov Mixtures Process

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