Nonparametric Estimation of the Ruin Probability in the Classical Compound Poisson Risk Model

In this paper we study estimating ruin probability which is an important problem in insurance. Our work is developed upon the existing nonparametric estimation method for the ruin probability in the classical risk model, which employs the Fourier transform but requires smoothing on the density of the sizes of claims. We propose a nonparametric estimation approach which does not involve smoothing and thus is free of the bandwidth choice. Compared with the Fourier-transformation-based estimators, our estimators have simpler forms and thus are easier to calculate. We establish asymptotic distributions of our estimators, which allows us to consistently estimate the asymptotic variances of our estimators with the plug-in principle and enables interval estimates of the ruin probability.

Solution of Ruin Probability for Continuous Time Model Based on Block Trigonometric Exponential Neural Network

The ruin probability is used to determine the overall operating risk of an insurance company. Modeling risks through the characteristics of the historical data of an insurance business, such as premium income, dividends and reinvestments, can usually produce an integral differential equation that is satisfied by the ruin probability. However, the distribution function of the claim inter-arrival times is more complicated, which makes it difficult to find an analytical solution of the ruin probability. Therefore, based on the principles of artificial intelligence and machine learning, we propose a novel numerical method for solving the ruin probability equation. The initial asset u is used as the input vector and the ruin probability as the only output. A trigonometric exponential function is proposed as the projection mapping in the hidden layer, then a block trigonometric exponential neural network (BTENN) model with a symmetrical structure is established. Trial solution is set to meet the initial value condition, simultaneously, connection weights are optimized by solving a linear system using the extreme learning machine (ELM) algorithm. Three numerical experiments were carried out by Python. The results show that the BTENN model can obtain the approximate solution of the ruin probability under the classical risk model and the Erlang(2) risk model at any time point. Comparing with existing methods such as Legendre neural networks (LNN) and trigonometric neural networks (TNN), the proposed BTENN model has a higher stability and lower deviation, which proves that it is feasible and superior to use a BTENN model to estimate the ruin probability.

On a Periodic Capital Injection and Barrier Dividend Strategy in the Compound Poisson Risk Model

In this paper, we assume that the reserve level of an insurance company can only be observed at discrete time points, then a new risk model is proposed by introducing a periodic capital injection strategy and a barrier dividend strategy into the classical risk model. We derive the equations and the boundary conditions satisfied by the Gerber-Shiu function, the expected discounted capital injection function and the expected discounted dividend function by assuming that the observation interval and claim amount are exponentially distributed, respectively. Numerical examples are also given to further analyze the influence of relevant parameters on the actuarial function of the risk model.

On Periodic Dividends for the Classical Risk Model with Debit Interest

A periodic dividend problem is studied in this paper. We assume that dividend payments are made at a sequence of Poisson arrival times, and ruin is continuously monitored. First of all, three integro-differential equations for the expected discounted dividends are obtained. Then, we investigate the explicit expressions for the expected discounted dividends, and the optimal dividend barrier is given for exponential claims. A similar study on a generalized Gerber–Shiu function involving the absolute time is also performed. To demonstrate the existing results, we give some numerical examples.

A VaR-Type Risk Measure Derived from Cumulative Parisian Ruin for the Classical Risk Model

In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.