Wavelets method for computing finite time Gerber-Shiu function

Abstract

In this thesis, a wavelets scheme is proposed to compute the finite time Gerber-Shiu function under a Levy subordinator model. Gerber-Shiu function, also known as expected discounted penalty function, was first introduced in 1998 and has then become a focus of research. This is because we can compute from this function various risk quantities including time of ruin, ruin probability, surplus before ruin and deficit at ruin, hence a very important entity in risk theory. In insurance industry, the scope of Solvency II Directive encourages risk management practices that defines required capital to manage risk, therefore promoting financial stability of the sector. The main proposes of Solvency II are to reduce the risk that an insurer fails to meet claims, reduce losses suffered by policyholders if ruin occurs, and to provide early warning to supervisors if capital falls below required levels. All of these are related to the risk quantities that can be computed from the Gerber-Shiu function mentioned above. Notwithstanding the great applications of Gerber-Shiu function, its mathematical expression involves infinitely many convolutions. In most practical cases, evaluation of this function is very difficult, if not impossible. In this thesis, an efficient numerical method based on wavelets is proposed to tackle this computational intractability. Wavelets are function basis that possess nice properties such as multiresolution and widely used in signal processing and denoising. They also have applications in solving PDEs and option pricing due to their computational efficiency. Under a similar reason, our numerical scheme using wavelets to simplify the infinite convolutions in the Gerber-Shiu function results in a neat and robust method. Furthermore, our proposed method can handle both the traditional Gerber-Shiu function as well as the finite time Gerber-Shiu function, which becomes more and more important from a regulatory point of view, via a proper Laplace transform. Error estimate of our scheme has been obtained via the wavelets expansion error and quadrature error. Finally, numerical experiments are performed to illustrate the efficiency and accuracy of our scheme.