I have a very general question. I am reading this paper :
http://www.webmeets.com/files/papers/eaere/2015/177/Discounting-HelsinkiBlind.pdf
There is a catastrophic event probability and after the catastrophic event the consumption level is reduced to zero. However, authors are making a steady state analysis prior to the catastrophic event. The catastrophic event probability is $h\left(X\right)$ (let s say that it follows a Poisson process.) $X$ stands for the pollution for example.
Let $T$ be the event occurrence time and denote $F\left(t\right)=Pr\left\{ T\leq t\right\}$ and $f\left(t\right)=F^{'}\left(t\right)$ as the corresponding probability distribution and density functions respectively.
$$h\left(S\left(t\right)\right)\Delta=\frac{f\left(t\right)\Delta}{1-F\left(t\right)}=-\frac{d\left[ln\left(1-F\left(t\right)\right)\right]}{dt}$$
where $\Delta$ is an infinitesimal time interval. The term $h\left(S\left(t\right)\right)\Delta$ specifies the conditional probability that an abrupt event will occur between $\left[t,t+\Delta\right]$.
My question is that : Due to this specification, the probability distribution function will be equal to 1 when $t$ tends to $\infty$. Then, in this case, for sure that catastrophic event will occur at the long run.
So, how is it possible to discuss about a steady state with catastrophic events ? At some moment in time, the economy will change regime with the catastrophic event and this steady state will not be a "permanent" one. How can it be possible to justify this ?