Limit theorems for Cox–Ingersoll–Ross process with externally and self-exciting jumps and application to finance

In this paper, we discuss the Cox–Ingersoll–Ross (CIR) process which incorporates the external and self-exciting jumps having exponential exciting function. This is the generalization of the classical CIR process and CIR process with Poisson and Hawkes jumps and it belongs to the class of affine point processes. By employing the concept of infinitesimal generator, we obtain some limit theorems, such as the Law of Large Numbers and the Central Limit Theorem. Furthermore, we investigate the Large Deviation Principle by using the Feynman–Kac formula. The last section of this paper uses the proposed model to analyze the impact of defaults on financial risk and the corresponding loss processes.