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

• A Cox risk model with periodic capital injections and dividend payments is used to model the insurance company surplus level. • The stochastic intensity process is represented by a general stochastic differential equation. • The insurer discrete observes the level of surplus and makes capital injections, dividend payments, and ruin decisions based on the observed level of surplus. • The continuous time Markov chain technique is combined with the Fourier cosine series expansion method to compute the finite-time expected present value of operating costs until ruin. • The optimal critical levels of capital injections and dividend payments are investigated, as well as the impact of some parameters on the results. 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 b 2 > 0 , the excess amount is paid as a lump sum of dividends; whenever the observed surplus level is between zero and another critical level b 1 (0 < b 1 < b 2), capital is injected to the surplus process so that it return to the level b 1 ; 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. [ABSTRACT FROM AUTHOR]