In the real world, the historical performance of a stock may have impacts on its dynamics and this suggests us to consider models with delays. We consider some portfolio optimization problem of Merton’s type in which the risky asset is described by some stochastic delay models. By virtue of the dynamic programming principle, we derive the Hamilton-Jacobi-Bellman (HJB) equations, which turn out to be nonlinear degenerate partial differential equations. Despite the challenge caused by the nonlinearity and the degeneration, we establish some existence results and the verification results.