  In this paper, we study the portfolio optimization problem with general
utility functions and when the return and volatility of underlying asset are
slowly varying. An asymptotic optimal strategy is provided within a specific
class of admissible controls under this problem setup. Specifically, we first
establish a rigorous first order approximation of the value function associated
to a fixed zeroth order suboptimal trading strategy, which is given by the
heuristic argument in [J.-P. Fouque, R. Sircar and T. Zariphopoulou, {\it
Mathematical Finance}, 2016]. Then, we show that this zeroth order suboptimal
strategy is asymptotically optimal in a specific family of admissible trading
strategies. Finally, we show that our assumptions are satisfied by a particular
fully solvable model.
