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In this paper, we study a generalised CIR process with externally-exciting and self-exciting jumps, and focus on the distributional properties and applications of this process and its aggregated process. The aim of the paper is to introduce a more general process that includes many models in the literature with self-exciting and external-exciting jumps. The first and second moments of this jump-diffusion process are used to calculate the insurance premium based on mean-variance principle. The Laplace transform of aggregated process is derived, and this leads to an application for pricing default-free bonds which could capture the impacts of both exogenous and endogenous shocks. Illustrative numerical examples and comparisons with other models are also provided.

A Generalised CIR Process with Externally-Exciting and Self-Exciting Jumps and Its Applications in Insurance and Finance