De Finetti?s dividend problem has been fascinating mathematicians and economists for half a century. This presentation will center on my recent work, directed by Professor Edward Waymire, which will be established under the simplest assumption (no external financing or consumption). This particular model has been extensively studied in recent twenty years. We will show that if the accumulated interest rate is a class of Levy process, the optimal dividend strategy is constant barrier when the surplus process is a spectrally negative Levy process. The approriate sufficient condition will be given. In the talk, fluctuation theory for Levy process, the Hamilton Jacobi Belman equation and maximal inequality will be discussed.