Some ruin problems for the MAP risk model
Introduction
In this paper we consider a general class of risk models with claims arriving according to a Markovian arrival process (MAP) (see Neuts, 1979). We denote by the insurer’s surplus at time , which is given by where is the initial surplus and is the aggregate claim amount at time , with being the number of claims by time and being the amount of the th claim. We assume that the evolution of the insurer’s surplus is driven by an underlying irreducible continuous-time Markov chain on the state space , such that the premiums are collected linearly at rate while . The intensity matrix for is with , where represents the intensity of state changes without claim arrivals and represents the intensity of state changes with an accompanying claim, with for all and . Furthermore, claims arriving with a transition from state to state are assumed to be independent of all other claims, to have distribution function with density , mean and Laplace transform . For this model, the positive loading condition is where is the stationary distribution of with . This condition is assumed to hold throughout.
Now define to be the time of ruin, with if for all . Letting denote the indicator function, we define for and to be the Laplace transform of the time of ruin with ruin caused by a claim in environment state , given initial surplus and initial environment state . We remark that it follows from Feng and Shimizu (2014) that can be expressed in terms of the -scale function defined in Ivanovs and Palmowski (2012). Further, is the Laplace transform of the time of ruin, given initial surplus and initial environment state . In particular, when is denoted which is defined as So is the ultimate ruin probability with ruin caused by a claim in environment state , given that the initial surplus is and the initial environment state is . Hence is the probability of ultimate ruin given that the initial surplus is and the initial environment state is , and is the non-ruin probability given that the initial surplus is and the initial environment state is .
An introduction to the Markovian arrival process can be found in Neuts (1979) and Breuer (2002). Studies on ruin-related quantities and dividend problems for the perturbed and non-perturbed MAP risk model can be found in Badescu et al., 2005, Badescu et al., 2007, Badescu (2008), Cheung and Landriault (2009), Cheung and Feng (2013), Li and Ren (2013), Feng and Shimizu (2014) and references therein. Ren (2008) studies the Laplace transform of the aggregate claims over for the MAP risk model.
The MAP risk model is very broad and it contains the classical risk model with , the phase-type renewal risk model (Ren, 2007), the semi-Markovian risk model studied by Albrecher and Boxma (2005), as well as the Markov-modulated risk model proposed by Asmussen (1989) and further studied by a number of authors including Asmussen and Rolski (1994), Ng and Yang (2006), Li and Lu (2008), and references therein. In particular, the MAP risk model simplifies to the Markov-modulated risk model if , (so that ), and with being the intensity matrix for the Markov process , with .
This paper is set out as follows. In the next section we give some notation and basic results. We then consider the distribution of the time of ruin by using two approaches in Sections 3 The distribution of the time of ruin under a constant premium rate, 4 The density of the time of ruin with varying premium rates. The probability function of the number of claims by the time of ruin is given in Section 5, and moments of the time of ruin are considered in Section 6.
Section snippets
Notation and basic relationships
In this section we introduce some notation and state some basic results. We start with claim numbers, and define to be the probability that claims occur before time , with the initial state being and the state at time being . Then is the probability that claims occur up to time with the initial state being .
We next consider aggregate claims distributions. Define
The distribution of the time of ruin under a constant premium rate
Define to be the probability that ruin does not occur by time and that the environment state at time is , given that the initial environment state is . Then is the non-ruin probability by time given that the initial environment state is . Let denote the defective density of the time of ruin with ruin caused by a claim in environment state given that the initial environment state is
The density of the time of ruin with varying premium rates
In this section we derive an alternative expression for the density of the time of ruin using transform inversion, and we let the premium rate vary according to the environment state. We recall that is the Laplace transform of the time of ruin from initial surplus given that the initial environment state is .
Conditioning on the events that can occur over an infinitesimal interval, we can show that
The distribution of the number of claims by the time of ruin
We now consider the distribution of the number of claims until ruin. This problem has been studied by a number of authors including Egídio dos Reis (2002) and Dickson (2012) for the classical risk model, and Stanford et al. (2000) who consider certain renewal risk models. The distribution of the number of claims until ruin can be used as a proxy to the distribution of , although caution should be exercised, as discussed in Stanford et al. (2000). Landriault et al. (2011) show that in certain
The moments of the time of ruin
We now consider moments of the time of ruin. Define to be the th moment of the time of ruin with ruin caused by a claim in environment state , given that the initial surplus is and the initial environment state is . Further, is the th moment of the time of ruin, given that the initial surplus is and the initial environment state is . We note that Further, let
Acknowledgements
This paper has benefited from comments from referees, and from colleagues at the University of Melbourne.
References (29)
- et al.
On the discounted penalty function in a Markov–dependent risk model
Insurance Math. Econom.
(2005) - et al.
A unified analysis of claim costs up to ruin in a Markovian arrival process
Insurance Math. Econom.
(2013) The joint distribution of the time to ruin and the number of claims until ruin in the classical risk model
Insurance Math. Econom.
(2012)How many claims does it take to get ruined and recovered?
Insurance Math. Econom.
(2002)- et al.
Potential measures for spectrally negative Markov additive processes with applications in ruin theory
Insurance Math. Econom.
(2014) - et al.
Occupation densities in solving exit problems for Markov additive process with one-sided jumps
Stochastic Process. Appl.
(2012) - et al.
Joint density involving the time to ruin in the Sparre Andersen risk model under exponential assumptions
Insurance Math. Econom.
(2011) - et al.
The maximum severity of ruin in a perturbed risk process with Markovian arrivals
Statist. Probab. Lett.
(2013) - et al.
The moments of the time of ruin, the surplus before ruin, and the deficit at ruin
Insurance Math. Econom.
(2000) - et al.
On the joint distribution of surplus before ruin and after ruin under a Markovian regime switching model
Stochastic Process. Appl.
(2006)
Ruin probabilities based on claim instants for some non-Poisson claim processes
Insurance Math. Econom.
Risk theory in a Markovian environment
Scand. Actuar. J.
Risk theory in a periodic environment: Lundberg’s inequality and the Cramer–Lundberg approximation
Math. Oper. Res.
Discussion of ‘The discounted joint distribution of the surplus prior to ruin and the deficit at ruin in a Sparre Andersen model’
N. Am. Actuar. J.
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