Some applications of Fourier-cosine method in business

Abstract

Based on the Fourier-cosine method, an efficient approximation of the finite-time Gerber-Shiu function and that of the cost objective in a production-inventory management problem are discussed in this thesis.

In the first part, under a risk process driven by a generic L\'{e}vy subordinator, the first systematic numerical study on finite-time Gerber-Shiu functions is provided via the popular Fourier-cosine method. 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, with an arbitrary penalty, an explicit integral expression for these functions is derived, which is 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 L\'{e}vy 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. Finally, the effectiveness of our approximation method is illustrated through different representative numerical experiments, some of them, such as those driven by Gamma and Generalized Stable Processes, are even achieved for the first time in the literature.

In the second part, a production planning problem for a continuous-review single-product production-inventory system is studied. The objective is to obtain the optimal replenishment rate that minimizes the total discounted cost under a L\'evy subordinator demand and general costs of holding inventory and lost sales. Firstly, the Laplace transform of the cost objective is explicitly obtained in terms of the unique positive root of the Lundberg equation. Based on that, the Fourier-cosine method is applied to approximate the cost objective. The corresponding error analysis is rigorously and thoroughly developed with explicit error bounds. Next, a hybrid algorithm is proposed to ensure the establishment of the globally optimal replenishment rate. In some special cases of demand and costs of inventory and lost sales, closed-form solutions for the cost objective and the optimal replenishment rate are given. Finally, abundant numerical experiments are conducted to illustrate the efficiency and robustness of our method, and extract related managerial insight.