Discrete time ruin probability with Parisian delay (original) (raw)
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Binomial discrete time ruin probability with Parisian delay
In this paper we analyze discrete time Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time ζ > 0. We identify expressions for the ruin probabilities with finite and infinite-time horizon. We find also their asymptotics when reserves tends to infinity. Finally, we calculate these probabilities for a few explicit examples.
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In this paper, we extend the concept of ruin in risk theory to the Parisian type of ruin. For this to occur, the surplus process must fall below zero and stay negative for a continuous time interval of specified length. We obtain the probability of ruin in the infinite horizon for ...
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Journal of Applied Probability, 2011
In this paper we analyze the so-called Parisian ruin probability, which arises when the surplus process stays below 0 longer than a fixed amount of time ζ > 0. We focus on a general spectrally negative Lévy insurance risk process. For this class of processes, we derive an expression for the ruin probability in terms of quantities that can be calculated explicitly in many models. We find its Cramér-type and convolution-equivalent asymptotics when reserves tend to ∞. Finally, we analyze some explicit examples.
Parisian ruin for the dual risk process in discrete-time
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In this paper we consider the Parisian ruin probabilities for the dual risk model in a discrete-time setting. By exploiting the strong Markov property of the risk process we derive a recursive expression for the finite-time Parisian ruin probability, in terms of classic discrete-time dual ruin probabilities. Moreover, we obtain an explicit expression for the corresponding infinite-time Parisian ruin probability as a limiting case. In order to obtain more analytic results, we employ a conditioning argument and derive a new expression for the classic infinite-time ruin probability in the dual risk model and hence, an alternative form of the infinite-time Parisian ruin probability. Finally, we explore some interesting special cases, including the binomial/geometric model, and obtain a simple expression for the Parisian ruin probability of the gambler's ruin problem.
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In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distribution -- Pareto distribution. The ruin probability for finite-horizon 5 and 10 was determined from recurrence equations. Moreover for exponential distribution the upper bound of ruin probability by Lundberg adjustment coefficient is given. For Pareto distribution the adjustment coefficient does not exist, hence an asymptotic approximation of the ruin probability if an initial capital tends to infinity is given. Obtained numerical results are given as tables and they are illustrated as graphs.
Parisian Ruin with Erlang Delay and a Lower Bankruptcy Barrier
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Parisian ruin occurs once the surplus stays continuously below zero for a given period. We consider the spectrally negative Lévy risk process where ruin is declared either at the first time that the reserve stays continuously below zero for an exponentially or mixed Erlang distributed random variable, or once it reaches a given negative threshold. We consider the Laplace transform of the time to ruin and the Laplace transform of the time that the process is negative.
Discounted penalty function at Parisian ruin for Lévy insurance risk process
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In the setting of a Lévy insurance risk process, we present some results regarding the Parisian ruin problem which concerns the occurrence of an excursion below zero of duration bigger than a given threshold r. First, we give the joint Laplace transform of ruin-time and ruin-position (possibly killed at the first-passage time above a fixed level b), which generalises known results concerning Parisian ruin. This identity can be used to compute the expected discounted penalty function via Laplace inversion. Second, we obtain the q-potential measure of the process killed at Parisian ruin. The results have semi-explicit expressions in terms of the q-scale function and the distribution of the Lévy process.
Some results of ruin probability for the classical risk process
Journal of Applied Mathematics and Decision Sciences, 2003
The computation of ruin probability is an important problem in the collective risk theory. It has applications in the fields of insurance, actuarial science, and economics. Many mathematical models have been introduced to simulate business activities and ruin probability is studied based on these models. Two of these models are the classical risk model and the Cox model. In the classical model, the counting process is a Poisson process and in the Cox model, the counting process is a Cox process. Thorin (1973) studied the ruin probability based on the classical model with the assumption that random sequence followed theΓdistribution with density functionf(x)=x1β−1β1βΓ(1/β)e−xβ,x>0, whereβ>1. This paper studies the ruin probability of the classical model where the random sequence follows theΓdistribution with density functionf(x)=αnΓ(n)xn−1e−αx,x>0, whereα>0andn≥2is a positive integer. An intermediate general result is given and a complete solution is provided forn=2. Simu...