On a time-changed Lévy risk model with capital injections and periodic observation

Abstract

In this paper, an insurance company’s surplus process is modeled by a time-changed Lévy process, where the continuous-time change is realized by integrating the Cox–Ingersoll–Ross (CIR) process. Assume that the surplus level is monitored periodically with constant monitoring frequency, and the insurer makes decisions on capital injections based on the observed surplus levels. Given a critical level b (b>0), whenever the observed surplus level belongs to the interval [0,b), capitals are injected into the insurance company to bring the surplus level back to b; whenever the observed surplus level is below level 0, ruin is declared, and the surplus process is stopped. The finite-time expected present value of operating costs until ruin is studied. Utilizing the Fourier series expansion in cosine form, we derive the closed-form approximation formula for this function and analyze the approximation error. Finally, many numerical examples are given to demonstrate the effectiveness of our method, and the impacts of some parameters are also studied.