Six Bridges to .'s

six methods are included for calculating the probability of ultimate ruin, where the waiting times between claims independent and identically distributed; those distributions need not be exponential. The first three use Monte Carlo and the last three do not. The first Monte carlo method uses two-dimensional random sequences to generate a distribution of maximal aggregate losses and the connection between the maximal aggregate loss randon variable and the probability of ruin. The first Convolution method is similar to the first Monte Carlo method except that the distribution of maximal aggregate loss is generated by performing two-dimensional generalized numerical "convolutions" The second Monte Carlo method uses one-dimensional random sequences to obtain the probability of ruin, starting with a given initial surplus. The second Convolution method is similar to the second Monte Carlo method except that the probability of ruin is generated by one-dimensional numerical convolutions. The third Monte Carlo method uses the second Monte Carlo method starting with zero initial surplus, together with the fact that the maximal aggregate loss random variable has a compound geometric distribution. The third Convolution method uses the second Convolution method starting with zero initial surplus, together with the fact that the maximal aggregate loss randon variable has a compound geometric distribution. Some preliminary one-dimensional convolutions are performed to establish two parameters n and . , which are then used in any of the six methods. The appendixes introduce techniques to facilitate the performing of one- and two-dimensional regular and generalized numerical convolutions.