Estimating Ruin Probability in an Insurance Risk Model with Stochastic Premium Income Based on the CFS Method

This paper considers the estimation of ruin probability in an insurance risk model with stochastic premium income. We first show that the ruin probability can be approximated by the complex Fourier series (CFS) expansion method. Then, we construct a nonparametric estimator of the ruin probability and analyze its convergence. Numerical examples are also provided to show the efficiency of our method when the sample size is finite.

Affine Storage and Insurance Risk Models

The aim of this paper is to analyze a general class of storage processes, in which the rate at which the storage level increases or decreases is assumed to be an affine function of the current storage level, and, in addition, both upward and downward jumps are allowed. To do so, we first focus on a related insurance risk model, for which we determine the ruin probability at an exponentially distributed epoch jointly with the corresponding undershoot and overshoot, given that the capital level at time 0 is exponentially distributed as well. The obtained results for this insurance risk model can be translated in terms of two types of storage models, in one of those two cases by exploiting a duality relation. Many well-studied models are shown to be special cases of our insurance risk and storage models. We conclude by showing how the exponentiality assumptions can be greatly relaxed.

Asymptotically Normal Estimators of the Gerber-Shiu Function in Classical Insurance Risk Model

Nonparametric estimation of the Gerber-Shiu function is a popular topic in insurance risk theory. Zhang and Su (2018) proposed a novel method for estimating the Gerber-Shiu function in classical insurance risk model by Laguerre series expansion based on the claim number and claim sizes of sample. However, whether the estimators are asymptotically normal or not is unknown. In this paper, we give the details to verify the asymptotic normality of these estimators and present some simulation examples to support our result.

An Uncertain Alternating Renewal Insurance Risk Model

The claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. However, the claim process actually includes two processes, which are called the report process and the payment process, respectively. An alternative way is to describe the claim process by an uncertain alternating renewal reward process. Therefore, this paper proposes an insurance risk model under uncertain measure in which the claim process is supposed to be an alternating renewal reward process and the premium process is regarded as a renewal reward process. Then, the paper also gives the inverse uncertainty distribution of the insurance risk process. The expression of ruin index and the uncertainty distribution of the ruin time are derived which both have explicit expressions based on given uncertainty distributions. Finally, several examples are provided to illustrate the modeling ideas.

Asymptotic Behavior of Ruin Probabilities in an Insurance Risk Model with Quasi-Asymptotically Independent or Bivariate Regularly Varying-Tailed Main Claim and By-Claim

This paper considers a by-claim risk model under the asymptotical independence or asymptotical dependence structure between each main claim and its by-claim. In the presence of heavy-tailed main claims and by-claims, we derive some asymptotic behavior for ruin probabilities.

Estimating the Expected Discounted Penalty Function in a Compound Poisson Insurance Risk Model with Mixed Premium Income

In this paper, we consider an insurance risk model with mixed premium income, in which both constant premium income and stochastic premium income are considered. We assume that the stochastic premium income process follows a compound Poisson process and the premium sizes are exponentially distributed. A new method for estimating the expected discounted penalty function by Fourier-cosine series expansion is proposed. We show that the estimation is easily computed, and it has a fast convergence rate. Some numerical examples are also provided to show the good properties of the estimation when the sample size is finite.