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

Yunyun Wang,Wenguang Yu,Hongli Fan,Yujuan Huang,Xinliang Yu

doi:10.3390/math7030305

Yunyun Wang, Wenguang Yu + Show 3 more

Open Access

https://doi.org/10.3390/math7030305

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Abstract

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.

Highlights

In this paper, we consider an insurance risk model with mixed premium income defined by M(t) U (t) = u + ct + ∑ i =1 N (t) Yi − Xj, t ≥ 0, (1)j =1 where u ≥ 0 is the initial surplus, c ≥ 0 is the constant premium rate, and U (t) denotes the surplus level of an insurance company at time t
Yang et al [66] used a two-dimensional COS method to estimate the discounted density function of the deficit at ruin in the classical risk model with stochastic income where the premiums are only described by a compound
As for the expected discounted penalty function, it follows from Equation (5) that for 0 ≤ u ≤ a, Φ(u) ≈ ΦK,a (u) :=

Summary

Introduction

We consider an insurance risk model with mixed premium income defined by. Bao [28], Bao and Ye [29] and Yang and Zhang [30] studied the Gerber-Shiu function in the classical risk model, delayed renewal risk model and Sparre Andersen risk model by assuming the premium process are Poisson process, respectively. Zhang [65] applied the COS method to compute the density of the time to ruin in the classical risk model. Yang et al [66] used a two-dimensional COS method to estimate the discounted density function of the deficit at ruin in the classical risk model with stochastic income where the premiums are only described by a compound. We use the COS method to estimate the expected discounted penalty function in this risk model with mixed premiums income process.

Fourier-Cosine Series Expansion

The Fourier Transform of Expected Discounted Penalty Function

Estimation Procedure

Simulation Studies