A consumption-investment problem with constraints on minimum and maximum consumption rates

We develop famous Merton's financial model to more realistic and interesting situation: consumption rate has lower and upper constraints. The aim is to find optimal strategies for consumption and investment. The corresponding HJB equation is a fully nonlinear ordinary differential equation. We use stochastic analysis and differential equation technique to find optimal strategies. The regularity of the value function is obtained as well. The method and result are meaningful and interesting in both of finance and mathematics.