Insurance with borrowing: first- and second-order approximations

Abstract

We consider the operation of an insurer with a large initial surplus x>0$. The insurer's surplus process $S(t)$ (with S(0)=x) evolves in the range S(t)\geq 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T₁,T₂,.... At the time T_\eta(x) when the process S(t) first becomes negative, the insurer's ruin (in the `classical' sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some `critical' level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x࢐∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(T_η(x)).