  Empirical studies indicate the presence of multi-scales in the volatility of
underlying assets: a fast-scale on the order of days and a slow-scale on the
order of months. In our previous works, we have studied the portfolio
optimization problem in a Markovian setting under each single scale, the slow
one in [Fouque and Hu, SIAM J. Control Optim., 55 (2017), 1990-2023], and the
fast one in [Hu, Proceedings of IEEE CDC 2018, accepted]. This paper is
dedicated to the analysis when the two scales coexist in a Markovian setting.
We study the terminal wealth utility maximization problem when the volatility
is driven by both fast- and slow-scale factors. We first propose a zeroth-order
strategy, and rigorously establish the first order approximation of the
associated problem value. This is done by analyzing the corresponding linear
partial differential equation (PDE) via regular and singular perturbation
techniques, as in the single-scale cases. Then, we show the asymptotic
optimality of our proposed strategy within a specific family of admissible
controls. Interestingly, we highlight that a pure PDE approach does not work in
the multi-scale case and, instead, we use the so-called epsilon-martingale
decomposition. This completes the analysis of portfolio optimization in both
fast mean-reverting and slowly-varying Markovian stochastic environments.
