Application of Stochastic Differential Equations to Reserving in Non-Life Insurance

This article applies stochastic differential equations (SDEs), the Cox–Ingersoll–Ross (CIR) model and arithmetic Brownian motion (ABM), to non-life insurance reserving. Following Dal Moro (2021), the chain ladder method is shown equivalent to a CIR model under suitable parameter choices. Similarly, drawing on Mack (2008) and Saluz (2015), the Bornhuetter–Ferguson and Cape Cod methods are shown equivalent to an ABM. These models provide robust frameworks for reserve dynamics while requiring minimal assumptions and clarify the statistical distribution of reserving risk. While reserving risk is typically assumed lognormal, the CIR and ABM models yield alternative distributions: the chain ladder model produces a noncentral chi-square distribution with zero degrees of freedom, whereas the Bornhuetter–Ferguson model yields a normal distribution. The study highlights the theoretical foundations, practical implementation, and benefits of SDEs in non-life reserving, particularly for selecting a distribution to model overall reserving risk.