The Expected Value of Discounted Penalty at Ruin Time

Penalty function is the function of the dificit at ruin and the surplus immediately before ruin, and the previous work mainly studied the expectation of discounted penalty in the case of constant force of interest. In this paper, we make a further study on the expectation of discounted penalty and obtain the renewal equation of the expectation of discounted penalty in the case of force of interest with Poisson jump. Some results in classical risk theory are also discussed by virtue of this equation.