On the duration of negative surplus

Doutoramento em Matemática

In ruin theory, assuming the classical compound Poisson continuous time surplus process, we consider the process as continuing if ruin occurs. Due to the assumptions presented, the surplus will drift to infinity with probability one. If ruin occurs, the process will stay temporarily below zero and will recover to positive values with probability one. Using a martingale method introduced by Gerber [1990], we derive the moment generating function of the duration of a period of negative surplus or the time to recovery, given that ruin occurs. An explicit expression for this depends on the availability of the moment generating function of the conditional severity of ruin, which is not available in general. However, we show that in some special cases we can get explicit formulae. When the initial surplus is equal to zero we also show that the duration of a period of negative surplus has the same distribution as the time to ruin. We present asymptotic formulae for the moment generating function of the con­ditional time to recovery, as well as for some other functions in ruin theory. For the situations where explicit formulae do not exist, we developed an algo­ rithm based on the discrete time model presented by Dickson and Waters [1991], in order to calculate approximate values for the moments of both the severity of ruin and the time to recovery. Since the surplus can fall below zero more than once, we present formulae for the distribution and moment generating function of the number of periods of negative surplus, as well as the moment generating function of the total duration of negative surplus. Following the work of Gerber [1990] and the developments above, we present formulae for the moment generating function for both the conditional number of claims occurring during a single period of negative surplus, given that the surplus falls below zero, and the total number of claims during periods of negative surplus, allowing for the possibility of the surplus falling below zero more than once. We further study the effect of two reinsurance arrangements on the model, pro­ portional and excess of loss, showing that, when the initial surplus is zero the pro­ portional treaty gives greater conditional expected value and variance of the time to recovery and the time to ruin, given that ruin occurs, and that this does not neces­ sarily happen with the excess of loss treaty. For the excess of loss treaty we adapted the algorithm mentioned above to approximate the moments of the duration of a single period of negative surplus, for a positive initial surplus. Furthermore, following the method of Dickson [1992] we give an interpretation of some results in ruin theory by means of dual events.