Approach to self-similarity in Smoluchowski's coagulation equations

We consider the approach to self-similarity (or dynamical scaling) in Smoluchowski's equations of coagulation for the solvable kernels $K(x,y)=2$, $x+y$ and $xy$. In addition to the known self-similar solutions with exponential tails, there are one-parameter families of solutions with algebraic decay, whose form is related to heavy-tailed distributions well-known in probability theory. For K=2 the size distribution is Mittag-Leffler, and for $K=x+y$ and $K=xy$ it is a power-law rescaling of a maximally skewed $\alpha$-stable Levy distribution. We characterize completely the domains of attraction of all self-similar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.Comment: Latex2e, 42 pages with 1 figur

On approximation of functions satisfying defective renewal equations

Functions satisfying a defective renewal equation arise commonly in applied probability models. Usually these functions don't admit a explicit expression. In this work we consider to approximate them by means of a gamma-type operator given in terms of the Laplace transform of the initial function. We investigate which conditions on the initial parameters of the renewal equation give optimal order of uniform convergence in the approximation. We apply our results to ruin probability in the classical risk model, paying special attention to mixtures of gamma claim amounts

Modelling and Optimizing Imperfect Maintenance of Coatings on Steel Structures

Steel structures such as bridges, tanks and pylons are exposed to outdoor weathering conditions. In order to prevent them from corrosion they are protected by an organic coating system. Unfortunately, the coating system itself is also subject to deterioration. Imperfect maintenance actions such as spot repair and repainting can be done to extend the lifetime of the coating. In this paper we consider the problem of finding the set of actions that minimizes the expected maintenance costs over a bounded horizon. To this end we model the size of the area affected by corrosion by a non-stationary gamma process. An imperfect maintenance action is to be done as soon as a fixed threshold is exceeded. The direct effect of such an action on the condition of the coating is assumed to be random. On the other hand, maintenance may also change the parameters of the gamma deterioration process. It is shown that the optimal maintenance decisions related to this problem are a solution of a continuous-time renewal-type dynamic programming equation. To solve this equation time is discretized and it is verified theoretically that this discretization induces only a small error. Finally, the model is illustrated with a numerical example.non-stationary gamma process;condition-based maintenance;degradation modelling;imperfect maintenance;life-cycle management;renewal-type dynamic programming equation

Approximations to ruin probablities in infinite time using a Lévy process

Mestrado em Ciências ActuariaisEsta dissertação aborda especificamente problemas da área da teoria da ruína, sub-área da teoria do risco para a atividade seguradora. Em particular, estudamos a probabilidade de ruína eventual. Adaptamos o modelo de risco coletivo de Cramér-Lundberg, estendendo para o modelo perturbado. Adicionamos ao modelo de Poisson composto uma componente representativa de um processo de Lévy (alfa estável). Esta componente adicional permite-nos incorporar incertezas decorrentes de, por exemplo, flutuações de taxas de juro, alterações no número de apólices na carteira, em quaisquer dos casos mantendo as hipóteses tradicionais. Com o objetivo de cálculo da probabilidade de ruína no modelo perturbado, apresentamos novas técnicas, recuperando e generalizando modelos de aproximação bem conhecidos, tais como os de DE VYLDER (1996), DUFRESNE AND GERBER (1989), POLLACZEK-KHINCHINE, PADÉ (ver AVRAM ET AL. (2001) e JOHNSON AND TAAFFE (1989)), obtidas ajustando um, dois, três ou quatro momentos ordinários da distribuição dos montantes das indemnizações. Para além disso, considerámos também importante que as aproximações ajustassem a transformada de Laplace (para a probabilidade de ruína), veja-se FURRER (1998). Avaliamos a qualidade das aproximações estudadas exemplificando para um conjunto de distribuições de cauda leve e de cauda pesada. Ilustramos com detalhe com alguns resultados numéricos.In this thesis, we work with prominence to a key area in actuarial science, namely ruin theory. The Cramér-Lundberg model of collective risk theory is adapted for the perturbed model, by adding a Lévy (α-stabled) process to the compound Poisson process, which allows us to consider uncertainty to the premium income, fluctuations of the interest rates, changes to the number of policyholders, without neglecting all other assumptions. On the way, we present new approximation techniques, built for the perturbed model in infinite time, and recall a remarkable family of well-known approximations by DE VYLDER (1996), DUFRESNE AND GERBER (1989), POLLACZEK-KHINCHINE and PADÉ (see AVRAM ET. AL (2001) and JOHNSON AND TAAFFE (1989)), obtained by fitting one, two, three or four (we also attempt five) ordinary moments of the claim amount distribution, and thus significantly generalising these approximations. Finding such approximation which fit the Laplace transform of the ruin probability would also be quite valuable, see FURRER (1998). We test the accuracy of the approximations using a mixture of light and heavy tailed distributions for the individual claim amount. We evaluate the ultimate ruin probability and illustrate in detail some numerical results.N/

Randomized observation times for the compound Poisson risk model: The discounted penalty function

In the framework of collective risk theory, we consider a compound Poisson risk model for the surplus process where the process (and hence ruin) can only be observed at random observation times. For Erlang(n) distributed inter-observation times, explicit expressions for the discounted penalty function at ruin are derived. The resulting model contains both the usual continuous-time and the discrete-time risk model as limiting cases, and can be used as an effective approximation scheme for the latter. Numerical examples are given that illustrate the effect of random observation times on various ruin-related quantities