Optimal dividends under Erlang(2) inter-dividend decision times with nonlinear surplus

In the classic dividend surplus model, initial surplus are generally fixed and premiums or incomes increased linearly. In our research, we consider that the initial surplus and income in the Erlang(2) dividend model will also change over time. This model contains both the continuous-time and the discrete-time risk model as a limit and represents a certain type of bridge between them which still enables the explicit calculation of moments of total discounted dividend payments until ruin. In this paper, we adopt Statistical methods to convert the differential equations with variable coefficients to ordinary differential equation and further study the optimal periodic barrier strategies when the initial surplus is dynamic and the income is nonlinear under the condition of the inter-dividend decision times follow Erlang(2) distribution