On dividends and Gerber-Shiu analysis with constant interest and a periodic-threshold mixed strategy

In recent years, the research focus in insurance risk theory has shifted towards multi-type mixed dividend strategies. However, the practical factors and constraints in financial market transactions, such as interest rates, tax rates, and transaction fees, inevitably impact these strategies. By incorporating appropriate constraints, a multi-type mixed strategy can better simulate real-world transactions. Following the approach of Liu et al. [28], we examine a classical compound Poisson risk model that incorporates the constraints of constant interest rates and a periodic-threshold mixed dividend strategy. In this model, the surplus process of insurance companies is influenced by several factors. These factors include constant interest rates, continuously distributed dividends within intervals (threshold dividend strategy), and dividends at discrete time points (periodic dividend strategy). We derive the piecewise integro-differential equations (IDEs) that describe the expected present value of dividends (EPVDs) until ruin time and the Gerber-Shiu expected discounted penalty function. Furthermore, we provide explicit solutions to these IDEs using an alternative method based on the inverse Laplace transform combined with the Dickson-Hipp operator. This enables us to obtain explicit expressions for the dividend and Gerber-Shiu functions. Additionally, we present examples to illustrate the application of our results.