On ruin for the Erlang(n) risk process

A defective renewal equation is derived for the expected discounted penalty due at ruin, φδ(u)=E[e−δTw(U(T−),|U(T)|)I(T<∞)|U(0)=u],in a risk model with Erlang(n) claim inter-arrival times. The approach used is similar to that of Gerber and Shiu (1998), repeatedly integrating an nth order integro-differential equation for φ, n times. The joint distribution of three random variables is obtained, the ruin time, the surplus just before ruin and the deficit at ruin.