Ruin probabilities prepared for numerical calculation

Ever since Filip Lundberg in the beginning of this century (1903) founded the collective theory of risk the notion of probability of ruin has more or less explicitly played a fundamental role in the said theory. Considering an insurance portfolio we know that claims occur from time to time causing liabilities the discounted value of which (the amount of the claim) we assume to be determinable at the time epoch of the claim. Moreover, we assume that we have a probabilistic model for the set of possible realizations of the development of the portfolio, with other words we assume that we have a probabilistic model for the process of the portfolio. If the initial reserve u ⩾ 0 is attributed to the portfolio at time 0 the risk reserve at time t is a random variable obtained by adding to u the premiums corresponding to the time interval (0, t] after subtracting the amounts of claims occurred during the said interval. The ruin probability for the time interval (0, t] is now the probability that the risk reserve be negative at some time during the interval. This notion of ruin probability is also possible when we consider an unlimited interval t ⩾ 0. It is customary to use the notation Ψ(u, t) for the ruin probability for the time interval (0, t]. In case of an unlimited interval the corresponding notation is Ψ(u) Ψ(u, ∞).