On the Time Value of Absolute Ruin with Debit Interest

Assume that the surplus of an insurer follows a compound Poisson surplus process. When the surplus is below zero or the insurer is on deficit, the insurer could borrow money at a debit interest rate to pay claims. Meanwhile, the insurer will repay debts from her premium income. The negative surplus may return to a positive level if debts are reasonable. However, when the negative surplus is below some critical level, the surplus is no longer able to be positive. Absolute ruin is said to occur at this moment. Many questions about absolute ruin have not yet been solved. In this paper, we study absolute ruin questions by defining an expected discounted penalty function at absolute ruin or the Gerber-Shiu function at absolute ruin. The function includes the absolute ruin probability, the Laplace transform of the time to absolute ruin, the deficit at absolute ruin, the surplus just before absolute ruin, and many other quantities related to absolute ruin. First, we derive a system of integrodifferential equations satisfied by the Gerber-Shiu function and obtain a defective renewal equation that links the integro-differential equations in the system. Then, we show that when the initial surplus goes to infinity, the absolute ruin probability and the classical ruin probability are asymptotically equal for heavy-tailed claims while the ratio of the absolute ruin probability to the classical ruin probability goes to a positive constant that is less than one for light-tailed claims. Finally, we give explicit expressions for the Gerber-Shiu function at absolute ruin for exponential claims.