On finite-time ruin probabilities with reinsurance cycles influenced by large claims

Market cycles play a great role in reinsurance. Cycle transitions are not independent from the claim arrival process : a large claim or a high number of claims may accelerate cycle transitions. To take this into account, a semi-Markovian risk model is proposed and analyzed. A refined Erlangization method is developed to compute the finite-time ruin probability of a reinsurance company. As this model needs the claim amounts to be Phase-type distributed, we explain how to fit mixtures of Erlang distributions to long-tailed distributions. Numerical applications and comparisons to results obtained from simulation methods are given. The impact of dependency between claim amounts and phase changes is studied.

On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

The Distribution of the Discounted Compound PH-Renewal Process

The family of phase–type (PH) distributions has many useful properties such as closure under convolution and mixtures, as well as rational Laplace transforms. PH distributions are widely used in applications of stochastic models such as in queueing systems, biostatistics and engineering. They are also applied to insurance risk, such as in ruin theory. In this thesis, we extend the work of Wang (2007), that discussed the moment generating function (mgf) of discounted compound sums with PH inter–arrival times under a net interest δ ̸= 0. Here we focus on the distribution of the discounted compound sums. This represents a generalization of the classical risk model for which δ = 0. A differential equation system is derived for the mgf of a discounted compound sum with PH inter–arrival times and any claim severity if its mgf exists. For some PH inter–arrival times, we can further simplify this differential equation system. If inter–arrival times have a PH distribution of order 2, then second–order homogeneous differential equations are developed. By inverting the corresponding Laplace transforms, the extended density functions and cumulative distribution functions are also obtained. In addition, the series and transformation methods for solving differential equations is proposed, when the mean of inter–arrival times is small. Applications such as stop–loss premiums, and risk measures such as VaR and CTE are investigated. These are compared for different inter–arrival times. Some numerical examples are given to illustrate the results. Finally, asymptotical results are discussed, when the mean inter–arrival time goes to zero. For a fixed time, the asymptotic normal distribution is derived for discounted compound renewal sums

Implicit renewal theory for exponential functionals of L\'evy processes

We establish a new functional relation for the probability density function of the exponential functional of a L\'evy process, which allows to significantly simplify the techniques commonly used in the study of these random variables and hence provide quick proofs of known results, derive new results, as well as sharpening known estimates for the distribution. We apply this formula to provide another look to the Wiener-Hopf type factorisation for exponential functionals obtained in a series of papers by Pardo, Patie and Savov, derive new identities in law, and to describe the behaviour of the tail distribution at infinity and of the distribution at zero in a rather large set of situations

Linear predictor of discounted aggregated cash flows with dependent inter-occurrence time

Abstract : In this minor dissertation we derive the first two moments and a linear predictor of the compound discounted renewal aggregate cash flows when taking into account dependence within the inter-occurrence times. To illustrate our results, we use specific mixtures of exponential distributions to define the Archimedean copula, the dependence structure between the cash flow inter-occurrence times. The Ho-Lee interest rate model is used to show that the formulas derived can be calculated.M.Com. (Financial Economics