Finite-time expected present value of operating costs until ruin in a Cox risk model with periodic observation

Abstract

In this paper, we use a Cox risk model to describe the surplus flow of an insurance company, where the intensity process in the Cox process is assumed to follow a general stochastic differential equation. Suppose that the insurer observes the surplus process periodically with constant observation frequency. Whenever the observed surplus level is larger than a critical level b2>0, the excess amount is paid as a lump sum of dividends; whenever the observed surplus level is between zero and another critical level b1 (0<b1<b2), capital is injected to the surplus process so that it return to the level b1; whenever the observed surplus level is less than zero, ruin is declared and the process is stopped. Under these assumptions, we study the finite-time expected present value of operating costs until ruin. The continuous time Markov chain (CTMC) approximation technique is used to approximate the intensity process in the Cox model, and the Fourier cosine series expansion (COS) method is applied to approximate the function of interest. A lot of numerical results are given to show accuracy and efficiency of our method, and impacts of some parameters are also analyzed.