In this paper, we explore potential surplus modelling improvements by investigating how well the available models describe an insurance risk process. To this end, we obtain and analyse a real-life data set that is provided by an anonymous insurer. Based on our analysis, we discover that both the purchasing process and the corresponding claim process have seasonal fluctuations. Some special events, such as public holidays, also have impact on these processes. In the existing literature, the seasonality is often stressed in the claim process, while the cash inflow usually assumes simple forms. We further suggest a possible way of modelling the dependence between these two processes. A preliminary analysis of the impact of these patterns on the surplus process is also conducted. As a result, we propose a surplus process model which utilises a non-homogeneous Poisson process for premium counts and a Cox process for claim counts that reflect the specific features of the data.

In this paper we consider an extension of the Sparre Andersen insurance risk model by relaxing one of its independence assumptions. The newly proposed dependence structure is introduced through the premise that the joint distribution of the interclaim time and the subsequent claim size is bivariate phase-type (see, e.g. Assaf et al. (1984) and Kulkarni (1989)). Relying on the existing connection between risk processes and fluid flows (see, e.g. Badescu et al. (2005), Badescu, Drekic and Landriault (2007), Ramaswami (2006), and Ahn, Badescu and Ramaswami (2007)), we construct an analytically tractable fluid flow that leads to the analysis of various ruin-related quantities in the aforementioned risk model. Using matrix-analytic methods, we obtain an explicit expression for the Gerber–Shiu discounted penalty function (see Gerber and Shiu (1998)) when the penalty function depends on the deficit at ruin only. Finally, we investigate how some ruin-related quantities involving the surplus immediately prior to ruin can also be analyzed via our fluid flow methodology.