An insurance risk process with a generalized income process: A solvency analysis

In ruin theory, an insurer’s income process is usually assumed to grow at a deterministic rate of c>0 over time. For instance, both the well-known Cramér–Lundberg risk process and the Sparre Andersen risk model have this assumption built in the construction of their respective surplus processes. This assumption is mainly considered for purposes of mathematical tractability, but generally fails to accurately model an insurer’s income dynamics. To better characterize the variability and uncertainty of an insurer’s income process, several papers have studied insurance risk models with random incomes where the main emphasis is placed on carrying the related Gerber–Shiu analysis. However, a systematic and quantitative understanding of how the more volatile income processes impact an insurer’s solvency risk is still lacking. This paper aims to fill this gap in the literature by quantitatively assessing the impact of the choice of income process on some finite-time and infinite-time ruin quantities. To carry this analysis, we consider a generalized Sparre Andersen risk model with a random income process which renews at claim instants. For exponentially distributed claim sizes, we derive explicit expressions for some joint distributions involving the time to ruin and the number of claims until ruin. As special cases of the proposed insurance risk process, we consider income processes modelled by a subordinator or a particular varying premium rate model. Numerical examples are then carried to draw important risk management implications of a solvency nature for the insurer.