On the decomposition of the absolute ruin probability in a perturbed compound Poisson surplus process with debit interest

We consider a compound Poisson surplus process perturbed by diffusion with
debit interest. When the surplus is below zero or the company is on deficit, the company is
allowed to borrow money at a debit interest rate to continue its business as long as its debt
is at a reasonable level. When the surplus of a company is below a certain critical level, the
company is no longer profitable, we say that absolute ruin occurs at this situation. In this
risk model, absolute ruin may be caused by a claim or by oscillation. Thus, the absolute ruin
probability in the model is decomposed as the sum of two absolute ruin probabilities, where
one is the probability that absolute ruin is caused by a claim and the other is the probability
that absolute ruin is caused by oscillation.
In this paper, we first give the integro-differential equations satisfied by the absolute ruin
probabilities and then derive the defective renewal equations for the absolute ruin probabilities. Using these defective renewal equations, we derive the asymptotical forms of the
absolute ruin probabilities when the distributions of claim sizes are heavy-tailed and lighttailed. Finally, we derive explicit expressions for the absolute ruin probabilities when claim
sizes are exponentially distributed.