Extended Gerber-Shiu Expected Discounted Penalty Functions in Risk Model Perturbed by Diffusion and Application

An extension of the traditional risk surplus model is presented, integrating a diffusion process into the compound Poisson framework to enrich the model's descriptive power. This integration leads to the development of the Extended Gerber-Shiu (EGS) function, which introduces a premium for surpassing the initial capital level and takes into account two distinct stopping time scenarios. The problem formulation is derived using a retrospective differential analysis, resulting in a coupled integro-differential equation that the EGS function must satisfy. Addressing this formulation becomes the primary focus. The martingale approach is then applied, revealing that the EGS function can be partitioned into two simpler, independent problems: the fundamental Gerber-Shiu function and a first passage problem. The resolution of the latter is essential for the risk surplus model under study. Utilizing martingale measure transformation and Laplace transformation under various discount factors, the results that define the EGS function and its solution are determined. This work contributes to the field by enhancing the understanding and application of the EGS function in actuarial science and risk management, providing a foundation for further research and practical implementation.