The Fast Fourier Transform Algorithm in Ruin Theory for the Classical Risk Model

We focus on numerical evaluation of some quantities of interest in ruin theory, and on the practical use of the fast Fourier transform algorithm (FFT) in this context. We discuss the general application of the FFT for stochastic models, and we illustrate this by looking again at the probability of ruin in the classical risk model and by extending this approach to evaluation of the first moment of the time to ruin in the classical model. I. RUIN QUANTITIES IN THE CLASSICAL RISK MODEL In the classical risk model in insurance mathematics, claims arrive in a Poisson process, rate , and claim sizes are independent identically distributed random variables , independent of the claim arrivals process. Let denote the claim-size distribution function, and let , , assumed finite as necessary. Premiums arrive linearly in time at rate , and we write ! "# %$ & ' ( , where & , the premium loading factor, is assumed to be stricly positive. The surplus at time ) is *+",) ' .-/$0 1)32547698;:=< >@?BA > , where CED is the initial surplus, and F0",) ' is the number of claims that have arrived by time ) . Then ruin is said to occur if *+",) ' ever becomes negative. Let GH"I-J'K LNMPO QR)TS9*+",) ' UVDPW be the time of ruin. We have