School of Science, Anhui University of Science and Technology Department of Mathematics, Suzhou University

In this paper, we consider a risk model in
which the premium arriving number process is a Poisson process with
parameter $\lambda>0$ while the claim number process is the thinning
of the pre- mium arriving number process. Under such a model, we
obtain the integral equation, the integro- differential equation and
the recursive formula for the expected discounted penalty function.
Using the integro-differential equation we get the closed form
expressions for the Laplace transform of the time of ruin and the
deficit at ruin when the premium and the claim sizes are
exponentially distributed.