 Economics  Research Publications
Economics  Research Publications
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ItemThe density of the time to ruin for a Sparre Andersen process with Erlang arrivals and exponential claimsDickson, David C. M. ; Hughes, Barry D. ; Lianzeng, Zhang ( 200310)We derive expressions for the density of the time to ruin given that ruin occurs in a Sparre Andersen model in which individual claim amounts are exponentially distributed and interarrival times are distributed as Erlang(n, β). We provide numerical illustrations of finite time ruin probabilities, as well as illustrating features of the density functions.

ItemDe Vylder Approximations to the Moments and Distribution of the Time to RuinDickson, David C M ; Wong, Kwok Swan ( 200402)De Vylder (1978) proposed a method of approximating the probability of ultimate ruin in the classical risk model. In this paper we show that his ideas can be extended to approximate the moments and distribution of the time to ruin

ItemOptimal Dividends under a Ruin ProbabilityConstraintDickson, David CM ; Drekic, Steve ( 200510)We consider a classical surplus process modified by the paymentof dividends when the insurer’s surplus exceeds a threshold. We use aprobabilistic argument to obtain general expressions for the expected present value of dividend payments, and show how these expressions can be applied for certain individual claim amount distributions. We then consider the question of maximising the expected present valueof dividend payments subject to a constraint on the insurer’s ruin probability.

ItemOptimal dynamic reinsuranceDickson, David CM ; Waters, Howard R ( 200601)We consider a classical surplus process where the insurer can choosea different level of reinsurance at the start of each year. We assume theinsurer’s objective is to minimise the probability of ruin up to somegiven time horizon, either in discrete or continuous time. We developformulae for ruin probabilities under the optimal reinsurance strategy,i.e. the optimal retention each year as the surplus changes andthe period until the time horizon shortens. For our compound Poissonprocess, it is not feasible to evaluate these formulae, and hencedetermine the optimal strategies, in any but the simplest cases. Weshow how we can determine the optimal strategies by approximatingthe (compound Poisson) aggregate claims distributions by translatedgamma distributions, and, alternatively, by approximating the compoundPoisson process by a translated gamma process.

ItemThe GerberShiu discounted penalty function in the stationary renewal risk modelWillmot, Gordon E. ; Dickson, David C. M. ( 200208)The discounted penalty function introduced by Gerber and Shiu (1998) is considered in the stationary renewal risk model, where it is expressed in terms of the same discounted penalty function in the ordinary renewal risk model. This relationship unifies and generalizes known special cases. An invariance property between the stationary renewal risk model and the classical Poisson model with respect to the ruin probability is also generalized as a result.

ItemOn the expected discounted penalty function at ruin of a surplus process with interestCai, Jun ; Dickson, David C.M. ( 200111)In this paper, we study the expected value of a discounted penalty function at ruin of the classical surplus process modified by the inclusion of interest of the surplus. The 'penalty' is simply a function of the surplus immediately prior to ruin and the deficit at ruin. An integral equation for the expected value is derived, while the exact solution is given when the initial surplus is zero. Dickson's (1992) formulae for the distribution of the surplus immediately prior to ruin in the classical surplus process are generalised to our modified surplus process.

ItemRuin Probabilities with a Markov Chain Interest ModelCai, Jun ; Dickson, David C. M. ( 200208)Ruin probabilities in two generalized discrete time risk processes with a Markov chain interest model are studied. Recursive and integrat equations for the ruin probabilities are given. Generalized Lundberg inequalities for the ruin probabilities are derived both by inductive and martingale approaches. The relationships between these inequalities are discussed. A numerical example is given to illustrate these results.

ItemA Note on the Maximum Severity of Ruin and Related ProblemsDickson, David C. M. ( 200205)Picard (1994) defines the maximum severity of ruin to be the largest deficit of a classical surplus process. Starting from initial surplus Mu, between the time of ruin and the time of recovery to surplus level 0. He gives a simple expression for the distribution function of Mu in terms of the probability of ultimate ruin. This paper first addresses the question of calculating the moments of Mu. It is not easy to achieve explicit expressions for these despite knowing the distribution function of Mu. We consider situations where explicit expressions can be obtained,as well as approximations. We also consider the closely related question of the maximum surplus prior to ruin