On Ladder Height Distributions of General Risk Processes

We consider a continuous-time risk process {Ya(t);t.0} defined for a stationary marked point process {(Tn,Xn)}, where Ya(0)=a and Ya(t) increases linearly with a rate c and has a downward jump at time Tn with jump size Xn for n.{1,2,.}. For a=0, we prove that, under a balance condition, the descending ladder height distribution of {Y0(t)} has the same form as the case where {(Tn,Xn)} is a compound Poisson process. This generalizes the recent result of Frenz and Schmidt, in which the independence of {Tn} and {Xn} is assumed. In our proof, a differential equation is derived concerning the deficit Za at the ruin time of the risk process {Ya(t)} for an arbitrary a.0. It is shown that this differential equation is also useful for proving a continuity property of ladder height distributions.