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)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T1,T2,…. At the time Tη(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(Tη(x)).