On Ladder Height Distributions of General Risk Processes

Abstract

We consider a continuous-time risk process $\{Y_a(t); t \geq 0\}$ defined for a stationary marked point process $\{(T_n,X_n)\}$, where $Y_a(0) = a$ and $Y_a(t)$ increases linearly with a rate $c$ and has a downward jump at time $T_n$ with jump size $X_n$ for $n \in \{1,2,\ldots\}$. For $a = 0$, we prove that, under a balance condition, the descending ladder height distribution of $\{Y_0(t)\}$ has the same form as the case where $\{(T_n,X_n)\}$ is a compound Poisson process. This generalizes the recent result of Frenz and Schmidt, in which the independence of $\{T_n\}$ and $\{X_n\}$ is assumed. In our proof, a differential equation is derived concerning the deficit $Z_a$ at the ruin time of the risk process $\{Y_a(t)\}$ for an arbitrary $a \geq 0$. It is shown that this differential equation is also useful for proving a continuity property of ladder height distributions.