The optimal timing of investment under general Lévy process is described in some recent research papers. In this paper, firstly we classify investment decisions in four situations, and show a compact formula of optimal investment timing for each situation under Lévy process. Then we consider a double Erlang jump diffusion model, and derive the project value for an arbitrary investment threshold. By maximize the project value, the optimal investment timing and the corresponding value of project is obtained. Furthermore we show some results of sensitivity analysis for the optimal investment timing of double Erlang jump diffusion model, and describe several examples of usage of this model. Finally we derive equations for solving the entry-exit problem under double Erlang jump diffusion model.
