scholarly journals Estimating the Gerber-Shiu Expected Discounted Penalty Function for Lévy Risk Model

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Yujuan Huang ◽  
Wenguang Yu ◽  
Yu Pan ◽  
Chaoran Cui
Keyword(s):  
Statistical Estimation ◽  
Risk Model ◽  
Expansion Method ◽  
Expected Discounted Penalty Function ◽  
Surplus Process ◽  
Laguerre Series ◽  
Discounted Penalty Function ◽  
Simulation Results ◽  
Expected Discounted Penalty ◽  
Series Expansion Method

This paper studies the statistical estimation of the Gerber-Shiu discounted penalty functions in a general spectrally negative Lévy risk model. Suppose that the claims process and the surplus process can be observed at a sequence of discrete time points. Using the observed data, the Gerber-Shiu functions are estimated by the Laguerre series expansion method. Consistent properties are studied under the large sample setting, and simulation results are also presented when the sample size is finite.

The Markov Risk Model under Stochastic Discount Interest Force

2011 ◽  
Vol 179-180 ◽  
pp. 1080-1085
Author(s):  
Yu Juan Huang ◽  
Chun Ming Zhang
Keyword(s):  
Differential Equation ◽  
Penalty Function ◽  
Risk Model ◽  
Standard Wiener Process ◽  
Exponential Distributions ◽  
Expected Discounted Penalty Function ◽  
Geometric Process ◽  
Interest Force ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

We investigate the expected discounted penalty function in which the discount interest process is driven by markov process. We obtain the integro-differential equation satisfied by the expected discounted penalty function when interest process is perturbed by standard Wiener process and Poisson-Geometric process. A system of Laplace transforms of the expected discounted penalty function, given the initial environment state, is established from a system of integro-differential equations. One example is given with claim sizes that have exponential distributions.

scholarly journals Perturbed MAP Risk Models with Dividend Barrier Strategies

2009 ◽  
Vol 46 (2) ◽  
pp. 521-541 ◽  
Author(s):  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Arrival Process ◽  
Risk Models ◽  
Expected Discounted Penalty Function ◽  
Numerical Examples ◽  
Dividend Payments ◽  
Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.

scholarly journals Perturbed MAP Risk Models with Dividend Barrier Strategies

2009 ◽  
Vol 46 (02) ◽  
pp. 521-541 ◽  
Author(s):  
Eric C. K. Cheung ◽  
David Landriault
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Arrival Process ◽  
Risk Models ◽  
Expected Discounted Penalty Function ◽  
Numerical Examples ◽  
Dividend Payments ◽  
Discounted Penalty Function ◽  
Markov Modulated ◽  
Expected Discounted Penalty

In the context of a dividend barrier strategy (see, e.g. Lin, Willmot and Drekic (2003)) we analyze the moments of the discounted dividend payments and the expected discounted penalty function for surplus processes with claims arriving according to a Markovian arrival process (MAP). We show that a relationship similar to the dividend-penalty identity of Gerber, Lin and Yang (2006) can be established for the class of perturbed MAP surplus processes, extending in the process some results of Li and Lu (2008) for the Markov-modulated risk model. Also, we revisit the same ruin-related quantities in an identical MAP risk model with the only exception that the barrier level effective at time t depends on the state of the underlying environment at this time. Similar relationships are investigated and derived. Numerical examples are also considered.

The Expected Discounted Penalty Function with Random Income under Stochastic Discount Interest Force

2010 ◽  
Vol 113-116 ◽  
pp. 378-381
Author(s):  
Wen Guang Yu ◽  
Zhi Liu
Keyword(s):  
Poisson Process ◽  
Penalty Function ◽  
Risk Model ◽  
Classical Model ◽  
Expected Discounted Penalty Function ◽  
Renewal Model ◽  
Interest Force ◽  
Discounted Penalty Function ◽  
Premium Income ◽  
Expected Discounted Penalty

We study the delayed risk model with random premium income. The premium process is not a linear function of time in contrast with the classical model, but a Poisson process which is also independent of the claim process. We shall consider the case where the discount interest process is no longer a constant in comparison with the classical expected discounted penalty function, but a stochastic interest driven by Poisson process and Wiener process. The expected discounted penalty function in the delayed renewal model is expressed in terms of the corresponding Gerber-Shiu function in the ordinary renewal model. The obtained results can be viewed as the discrete analogy of the classical Sparre-Anderson risk model.

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.

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