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ItemThe moments of the time of ruin in Sparre Andersen risk modelsDickson, DCM (CAMBRIDGE UNIV PRESS, 20230301)Abstract We derive formulae for the moments of the time of ruin in both ordinary and modified Sparre Andersen risk models without specifying either the interclaim time distribution or the individual claim amount distribution. We illustrate the application of our results in the special case of exponentially distributed claims, as well as for the following ordinary models: the classical risk model, phasetype(2) risk models, and the Erlang( $\mathscr{n}$ ) risk model. We also show how the key quantities for modified models can be found.

ItemRuin problems in Markovmodulated risk modelsDickson, DCM ; Qazvini, M (Cambridge University Press, 20180301)Chen et al. (2014), studied a discrete semiMarkov risk model that covers existing risk models such as the compound binomial model and the compound Markov binomial model. We consider their model and build numerical algorithms that provide approximations to the probability of ultimate ruin and the probability and severity of ruin in a continuous time twostate Markovmodulated risk model. We then study the finite time ruin probability for a discrete mstate model and show how we can approximate the density of the time of ruin in a continuous time Markovmodulated model with more than two states.

ItemAn identity based on the generalised negative binomial distribution with applications in ruin theoryDickson, D (Cambridge University Press (CUP), 2019)In this study, we show how expressions for the probability of ultimate ruin can be obtained from the probability function of the time of ruin in a particular compound binomial risk model, and from the density of the time of ruin in a particular Sparre Andersen risk model. In each case evaluation of generalised binomial series is required, and the argument of each series has a common form. We evaluate these series by creating an identity based on the generalised negative binomial distribution. We also show how the same ideas apply to the probability function of the number of claims in a particular Sparre Andersen model.

ItemAnalysis of some ruinrelated quantities in a Markovmodulated risk modelLi, J ; Dickson, DCM ; Li, S (Taylor, 2016)In this paper, we study the joint Laplace transform and probability generating function of some random quantities that occur in each environment state by the time of ruin in a Markovmodulated risk process. These quantities include the duration spent in each state, the number of claims and the aggregate amount of claims that occurred in each state by the time of ruin. Explicit formulae for the joint transforms, given the initial surplus, and the initial and terminal environment states, are expressed in terms of a matrix version of the scale function. Moments and covariances of these ruinrelated quantities are obtained and numerical illustrations are presented. The joint transform of the duration spent in each state, the number of claims, and the aggregate amount of claims that occurred in each state by the time the surplus attains a certain level are also investigated.

ItemOptimal reinsurance under multiple attribute decision makingKARAGEYIK, BB ; DICKSON, DCM (Cambridge University Press (CUP), 2016)We apply methods from multiple attribute decision making (MADM) to the problem of selecting an optimal reinsurance level. In particular, we apply the Technique for Order of Preference by Similarity to Ideal Solution method with Mahalanobis distance. We consider the classical risk model under a reinsurance arrangement – either excess of loss or proportional – and we consider scenarios that have the same finite time ruin probability. For each of these scenarios we calculate three quantities: released capital, expected profit and expected utility of resulting wealth. Using these inputs, we apply MADM to find optimal retention levels. We compare and contrast our findings with those when decisions are based on a single attribute.

ItemGerberShiu analysis of a risk model with capital injectionsDICKSON, DCM ; Qazvini, M (Springer Verlag, 2016)We consider the risk model with capital injections studied by Nie et al. (Ann Actuar Sci 5:195–209, 2011; Scand Actuar J 2015:301–318, 2015). We construct a Gerber–Shiu function and show that whilst this tool is not efficient for finding the ultimate ruin probability, it provides an effective way of studying ruin related quantities in finite time. In particular, we find a general expression for the joint distribution of the time of ruin and the number of claims until ruin, and find an extension of Prabhu’s (Ann Math Stat 32:757–764, 1961) formula for the finite time survival probability in the classical risk model. We illustrate our results in the case of exponentially distributed claims and obtain some interesting identities. In particular, we generalise results from the classical risk model and prove the identity of two known formulae for that model.

ItemA note on some joint distribution functions involving the time of ruinDickson, DCM (Elsevier, 2016)In a recent paper, Willmot (2015) derived an expression for the joint distribution function of the time of ruin and the deficit at ruin in the classical risk model. We show how his approach can be applied to obtain a simpler expression, and by interpreting this expression by probabilistic reasoning we obtain solutions for more general risk models. We also discuss how some of Willmot’s results relate to existing literature on the probability and severity of ruin.

ItemThe maximum surplus before ruin in an Erlang(n) risk process and related problemsLI, S ; DICKSON, D ( 2006)

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
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