Approximations to ruin probability in the presence of an upper absorbing barrier

Abstract

Segerdahl (1970) considered a risk process with two absorbing barriers using methods which led to exact solutions for only particular claim amount distributions. The present paper develops an alternative method for exact solutions for ruin probabilities in the presence of an absorbing upper barrier and, by using an embedded Markov chain, obtains numerical approximations for such ruin probabilities for any claim amount distribution. Refinements to the approximating procedure are discussed. Comparison of exact ruin probabilities, when they can be identified, with the corresponding numerical approximations provides a measure of the accuracy of the method. Sensitivity of the method to changes in parameters is illustrated. When explicit exact solutions to ruin probabilities cannot be obtained, it appears likely that accurate approximations can be obtained by the Markov chain approach with less computational effort than is required to carry out numerical solution of the equation(s) for the exact value.

Some Optimal Dividends Problems

“…Consider first the case when 0 ≤ u < b. Then the probability that there is a first dividend stream is the probability that the surplus reaches b without ruin occurring first, the probability of which is x (u,b) where See Dickson and Gray (1984). Given that there is a first dividend stream, the probability that there is a second stream of dividend payments is p(b) where…”

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“…In the case that claims are phase type distributed with representation (θ, T ), we can arrive at an explicit expression for q u and q b . It is shown in Irbäck (2003), Dickson and Gray (1984a) that…”

“…Dickson and Gray (1984b) extended Segerdahl's (1970) results to the cases of gamma and hyper-exponential claim size distributions. In another paper, Dickson and Gray (1984a) developed an alternative method to approximate the ruin probability in the Segerdahl's (1970) model.…”

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“…He was the first to calculate the bankruptcy probability for a player who possesses initial capital u provided that this will happen no later than his capital reaches some level. Later, martingale methods of the analysis of random walks, Brownian motion with drift on a straight line allowed estimating the probability of bankruptcy on a finite time interval [1][2][3]. Gerber [4,5] was the first who applied martingale methods to estimate ruin probability.…”