Ruin Analysis for Erlang(2) Risk Process and American Put Option

In the classical risk theory, the risk of the accumulative claims follows Poission process. We will consider Erlang(2) risk process with the time between two claims following Erlang(2) distribution which always appears in control theory. In this paper,we consider an auxiliary function \phi(\cdot) which involves the time of ruin, the surplus immediately before ruin, and the deficit at the time of ruin for our model within the three variables are essential and principal for the study of risk process.This auxiliary function has been studied by Willmot and Lin (1999) in the classical continuous time risk model. Motivated by the exposition in Gerber and Shiu (1997)and Willmot and Lin (2000), the first important result is to find the joint distributiondensity function of U(T-) and |U(T)| which is convenient to get the expression of \phi(\cdot). But our approach is rather different from the technique for the classical risk model because of the distinct internal characteristic between two models. Influenced by the ideas in Gerber and Landry (1998) and Gerber and Shiu (1999), we will determine the optimal exercise price for an American put option whose foundation property price follows some risk process as an application.