scholarly journals The Gerber-Shiu Expected Penalty Function for the Risk Model with Dependence and a Constant Dividend Barrier

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Donghai Liu ◽  
Zaiming Liu ◽  
Dan Peng
Keyword(s):  
Linear Combination ◽  
Penalty Function ◽  
Integrodifferential Equation ◽  
Risk Model ◽  
Dependence Structure ◽  
Claim Amount ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Discounted Penalty Function ◽  
Constant Dividend Barrier

We consider a compound Poisson risk model with dependence and a constant dividend barrier. A dependence structure between the claim amount and the interclaim time is introduced through a Farlie-Gumbel-Morgenstern copula. An integrodifferential equation for the Gerber-Shiu discounted penalty function is derived. We also solve the integrodifferential equation and show that the solution is a linear combination of the Gerber-Shiu function with no barrier and the solution of an associated homogeneous integrodifferential equation.

scholarly journals A Note on a Generalized Gerber–Shiu Discounted Penalty Function for a Compound Poisson Risk Model

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 891 ◽  
Author(s):  
Jiechang Ruan ◽  
Wenguang Yu ◽  
Ke Song ◽  
Yihan Sun ◽  
Yujuan Huang ◽  
...  
Keyword(s):  
Laplace Transform ◽  
Exponential Distribution ◽  
Penalty Function ◽  
Risk Model ◽  
Recursive Method ◽  
Ruin Time ◽  
Compound Poisson ◽  
Compound Poisson Risk Model ◽  
Discounted Penalty Function ◽  
Poisson Risk Model

In this paper, we propose a new generalized Gerber–Shiu discounted penalty function for a compound Poisson risk model, which can be used to study the moments of the ruin time. First, by taking derivatives with respect to the original Gerber–Shiu discounted penalty function, we construct a relation between the original Gerber–Shiu discounted penalty function and our new generalized Gerber–Shiu discounted penalty function. Next, we use Laplace transform to derive a defective renewal equation for the generalized Gerber–Shiu discounted penalty function, and give a recursive method for solving the equation. Finally, when the claim amounts obey the exponential distribution, we give some explicit expressions for the generalized Gerber–Shiu discounted penalty function. Numerical illustrations are also given to study the effect of the parameters on the generalized Gerber–Shiu discounted penalty function.

scholarly journals On the Expected Discounted Penalty Function for the Classical Risk Model with Potentially Delayed Claims and Random Incomes

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Huiming Zhu ◽  
Ya Huang ◽  
Xiangqun Yang ◽  
Jieming Zhou
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Claim Amount ◽  
Classical Risk Model ◽  
Main Claim ◽  
Expected Discounted Penalty Function ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
The Classical Risk Model ◽  
Expected Discounted Penalty

We focus on the expected discounted penalty function of a compound Poisson risk model with random incomes and potentially delayed claims. It is assumed that each main claim will produce a byclaim with a certain probability and the occurrence of the byclaim may be delayed depending on associated main claim amount. In addition, the premium number process is assumed as a Poisson process. We derive the integral equation satisfied by the expected discounted penalty function. Given that the premium size is exponentially distributed, the explicit expression for the Laplace transform of the expected discounted penalty function is derived. Finally, for the exponential claim sizes, we present the explicit formula for the expected discounted penalty function.

scholarly journals Dependent Risk Models with Bivariate Phase-Type Distributions

2009 ◽  
Vol 46 (1) ◽  
pp. 113-131 ◽  
Author(s):  
Andrei L. Badescu ◽  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Fluid Flow ◽  
Penalty Function ◽  
Risk Model ◽  
Fluid Flows ◽  
Dependence Structure ◽  
Deficit At Ruin ◽  
Phase Type ◽  
The Deficit At Ruin ◽  
Discounted Penalty Function ◽  
Risk Processes

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.

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

Mathematics ◽  
2019 ◽  
Vol 7 (3) ◽  
pp. 305 ◽  
Author(s):  
Yunyun Wang ◽  
Wenguang Yu ◽  
Yujuan Huang ◽  
Xinliang Yu ◽  
Hongli Fan
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Insurance Risk ◽  
Compound Poisson ◽  
Expected Discounted Penalty Function ◽  
Fourier Cosine ◽  
Discounted Penalty Function ◽  
Premium Income ◽  
Expected Discounted Penalty ◽  
Insurance Risk Model

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.

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