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.

Ruin models with investment income

This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also the possibility of minimizing the ruin probability by investment and possibly reinsurance control. The main emphasis is on continuous time models, but discrete time models are also covered. A fairly extensive list of references is provided, particularly of papers published after 1998. For more references to papers published before that, the reader can consult [47].Comment: Published in at http://dx.doi.org/10.1214/08-PS134 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Approximations for time-dependent distributions in Markovian fluid models

In this paper we study the distribution of the level at time $\theta$ of Markovian fluid queues and Markovian continuous time random walks, the maximum (and minimum) level over $[0,\theta]$, and their joint distributions. We approximate $\theta$ by a random variable $T$ with Erlang distribution and we use an alternative way, with respect to the usual Laplace transform approach, to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration

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

International audienceMarket 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

The Markov Additive risk process under an Erlangized dividend barrier strategy

In this paper, we consider a Markov additive insurance risk process under a randomized dividend strategy in the spirit of Albrecher et al. (2011). Decisions on whether to pay dividends are only made at a sequence of dividend decision time points whose intervals are Erlang($n$) distributed. At a dividend decision time, if the surplus level is larger than a predetermined dividend barrier, then the excess is paid as a dividend as long as ruin has not occurred. In contrast to Albrecher et al. (2011), it is assumed that the event of ruin is monitored continuously (Avanzi et al. (2013) and Zhang (2014)), i.e. the surplus process is stopped immediately once it drops below zero. The quantities of our interest include the Gerber-Shiu expected discounted penalty function and the expected present value of dividends paid until ruin. Solutions are derived with the use of Markov renewal equations. Numerical examples are given, and the optimal dividend barrier is identified in some cases.postprin

On optimal dividend strategies in insurance with a random time horizon

For the classical compound Poisson surplus process of an insurance portfolio we investigate the problem of how to optimally pay out dividends to shareholders if the criterion is to maximize the expected discounted dividend payments until the time of ruin or a random time horizon, whichever is smaller. We explicitly solve this problem for an exponential time horizon and exponential claim sizes. Furthermore, we study the case of an Erlang(2) time horizon by introducing an external state process and derive the solution under the assumption that the external state process is observable. The results are illustrated by numerical examples

Asymptotic results for renewal risk models with risky investments

We consider a renewal jump-diffusion process, more specifically a renewal insurance risk model with investments in a stock whose price is modeled by a geometric Brownian motion. Using Laplace transforms and regular variation theory, we introduce a transparent and unifying analytic method for investigating the asymptotic behavior of ruin probabilities and related quantities, in models with light- or heavy-tailed jumps, whenever the distribution of the time between jumps has rational Laplace transform