The expression for the joint distribution of the surplus immediately before ruin and the deficit at ruin was expressed.

Under the assumption for existence of the adjustment coefficient,the asymptotic formulae for the ultimate ruin probability,the probabilities of surplus immediately before ruin and the deficit at ruin are given for sufficiently large initial surlus by means of a discrete key renewal limit theorem.

In this paper the joint distribution of the time of ruin,the surplus immediately before ruin,and the deficit at ruin is discussed in the compound binomial model.

The probability of ultimate ruin and the probability laws of the surplus immediately before ruin are discussed with emphasis.

Apart from ruin probability, the distribution of surplus immediately before ruin and that of deficit at ruin can also be used to characterize the risk of ruin in a insurance company In this paper, these distributions are studied with a renewal risk model under interest force; the recursive algorithms and integral equations for the distributions are obtained.

From the research,the recursion formulae of the distribution of the surplus immediately before ruin and of the distribution of the time in the red which describe the severity of ruin have been gotten.

Its purpose is to study the surplus prior to ruin and the distribution of the deficit at ruin for this risk process.