Computation of the probability of ultimate ruin and some other actuarial quantities under the classical risk model via Fast Fourier Transform

The Probability of ultimate ruin under the classical risk model is obtained as a solution of an integro -differential equation involving convolutions and we have used Fast Fourier Transform (FFT) to obtain the approximate values of the probability of ultimate ruin from this integro -differential equation under the situation when the claim severity is modelled by the Mixture of 3 Exponentials and the Weibull distribution. Another application of FFT in ruin theory is shown by means of applying it to obtain the quantiles of the aggregate claim distribution under these claim severity distributions. Extension of the application of FFT is shown by using it to obtain the first moment of the time to ruin under the classical risk model for these distributions. The distributions which have been used are such that one is light tailed and the another is heavy tailed so that a comparison can be made between them on the precision of the actuarial quantities obtained through FFT. FFT has been found to be efficient in obtaining these actuarial quantities when used in conjunction with certain modifications like exponential tilting to control the aliasing error.