On A Thinning Risk Model

We often suppose the premium arriving number process and the claim number process are independent in classical risk model. However, when insurance company manages it, the premium arriving number process is often correlated with the claim arriving process. For this, we consider the following risk model: the premium arriving number process is a Poisson process with a parameterλ, while the claim number process is a p -thinning process of the premium arriving number process.Expected discounted Penalty Function provides a unified method to study some ruin quantities and it is a hot topic of the ruin theory now. First, under the thinning risk model, we use the strong Markov property of the surplus process to derive the integral equation and the recursive formula for the expected discounted penalty function. Then when the premium and the claim sizes are exponentially distributed, we get the integro-differential equation for the expected discounted penalty function and the closed form expressions for the Laplace transform of the time of ruin, the deficit at ruin, the surplus immediately before ruin. At last, we give the analysis how the proportion of the claim number p affect the ruin probability.