Before revision:  We found that Merton's strategy that is optimal for continuous time model can be used effectively for the discrete market model that has sufficiently small time steps and approximate the continuous time model. After natural discretization, the Merton's strategy approximates the performance of the optimal strategy in  discrete time model .
After revision:  We found that Merton's strategy that is optimal for continuous time model can be used effectively for a discrete time market after a straightforward discretization. We found that Merton's strategy approximates the performance of the optimal strategy in discrete time model .