Fourier-Cosine Method for Finite-Time Gerber--Shiu Functions

Abstract

In this article, we provide the first systematic numerical study on, via the popular Fourier-cosine (COS) method, finite-time Gerber--Shiu functions with the risk process being driven by a generic Lévy subordinator. These functions play a major role in modern actuarial science, and there are still many open problems left behind such as the one here of looking for a universal effective numerical scheme for them. By extending the celebrated Ballot Theorem to the continuous setting, we first derive an explicit integral expression for these functions, with an arbitrary penalty, in terms of their infinite-time counterpart. As is common in actuarial or financial practice, an advanced knowledge of the characteristic function of the driving Lévy process facilitates the applicants of the Fourier-cosine method to this integral expression. Under some mild and practically feasible assumptions, a comprehensive and rigorous (yet demanding) error analysis is provided; indeed, up to an arbitrarily chosen error tolerance level, the numerical scheme is linear in computational complexity, which can even reach the theoretically fastest possible rate of 3; all of these are the most effective records of the contemporary state of the art in actuarial science. Finally, the effectiveness of our approximation method is illustrated through different representative numerical experiments, some of which, such as those driven by Gamma and Generalized Stable Processes, are even achieved for the first time in the literature, due to the limitations of most common existing approaches; we shall discuss this more in this article.