The Gerber-Shiu Discounted Penalty Function for Risk Process with Double Markovian Environment

2010 ◽  
Vol 108-111 ◽  
pp. 1103-1108
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Process ◽  
Closed Form Expression ◽  
Standard Wiener Process ◽  
Form Expression ◽  
Expected Discounted Penalty Function ◽  
Premium Rate ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Expected Discounted Penalty

In this paper, we study the Gerber-Shiu discounted penalty function. We shall consider the case where the discount interest process and the occurrence of the claims are driven by two distinguished Markov process, respectively. Moreover, in this model we also consider the influence of a premium rate which varies with the level of free reserves. Using backward differential argument, we derive the integral equation satisfied by the expected discounted penalty function via differential argument when interest process in every state is perturbed by standard Wiener process and Poisson process. In some special cases, closed form expression for these quantities are obtained.

On the Expected Discounted Penalty Function for a Risk Model with Thinning Process

2010 ◽  
Vol 29-32 ◽  
pp. 1156-1161
Author(s):  
Wen Guang Yu
Keyword(s):  
Penalty Function ◽  
Risk Model ◽  
Standard Wiener Process ◽  
Deficit At Ruin ◽  
Expected Discounted Penalty Function ◽  
Time Of Ruin ◽  
Special Cases ◽  
Discounted Penalty Function ◽  
Thinning Process ◽  
Expected Discounted Penalty

This paper studies the expected discounted penalty function for a risk model in which the arrival of insurance policies is a Poisson process and the process of claim occurring is -thinning process. Using backward differential argument, we derive the integro-differential equation satisfied by the expected discounted penalty function when the stochastic discount interest process is perturbed by standard Wiener process and Poisson-Geometric process. Applications of the integral equation are given to the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin occurs. In some special cases with exponential distributions, closed form expressions for these quantities are obtained.

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.

The Expected Discounted Penalty Function under Stochastic Discount Interest Force Driven by Markov Process

2012 ◽  
Vol 433-440 ◽  
pp. 5035-5039
Author(s):  
Chun Ming Zhang
Keyword(s):  
Differential Equation ◽  
Markov Process ◽  
Penalty Function ◽  
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 On a general class of renewal risk process: analysis of the Gerber-Shiu function

2005 ◽  
Vol 37 (03) ◽  
pp. 836-856 ◽  
Author(s):  
Shuanming Li ◽  
José Garrido
Keyword(s):  
Laplace Transform ◽  
Process Analysis ◽  
Risk Process ◽  
Expected Discounted Penalty Function ◽  
Renewal Equations ◽  
The Laplace Transform ◽  
Discounted Penalty Function ◽  
Renewal Risk ◽  
Defective Renewal Equations ◽  
Expected Discounted Penalty

We consider a compound renewal (Sparre Andersen) risk process with interclaim times that have a K n distribution (i.e. the Laplace transform of their density function is a ratio of two polynomials of degree at most n ∈ N). The Laplace transform of the expected discounted penalty function at ruin is derived. This leads to a generalization of the defective renewal equations given by Willmot (1999) and Gerber and Shiu (2005). Finally, explicit results are given for rationally distributed claim severities.

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