Stationary Distributions for Populations Subject to Random Catastrophes

Let X(t) denote the size of a certain population at time t **2267 0. It is assumed that at times when no catastrophes occur, the population grows according to the differential equation (1)$$ X'(t)=\alpha(X(t))$$. The hazard function for the occurrence of a catastrophe is β(X(t)), i.e. (2)$$\Pr\{no catastrophe occurs in the interval({T_{1}},{T_{2}})\}=\exp(-\int\limits_{{{T_{2}}}}^{{{T_{1}}}}{\beta(X(s))ds)}$$. If a catastrophe or downward jump takes place at time T, then it is assumed that (3)$$\Pr\{X(T)\leqq y|X(T-)=x\}=h(x,y)$$, where h is a given function. The problem will be to study the distribution of X(t), especially as t → ∞.