Title: Portfolio Optimization under Fast Mean-reverting and Rough Fractional Stochastic Environment,
Abstract: Fractional stochastic volatility models have been widely used to capture the
non-Markovian structure revealed from financial time series of realized
volatility. On the other hand, empirical studies have identified scales in
stock price volatility: both fast-time scale on the order of days and
slow-scale on the order of months. So, it is natural to study the portfolio
optimization problem under the effects of dependence behavior which we will
model by fractional Brownian motions with Hurst index $H$, and in the fast or
slow regimes characterized by small parameters $\eps$ or $\delta$. For the
slowly varying volatility with $H \in (0,1)$, it was shown that the first order
correction to the problem value contains two terms of order $\delta^H$, one
random component and one deterministic function of state processes, while for
the fast varying case with $H > \half$, the same form holds at order
$\eps^{1-H}$. This paper is dedicated to the remaining case of a fast-varying
rough environment ($H < \half$) which exhibits a different behavior. We show
that, in the expansion, only one deterministic term of order $\sqrt{\eps}$
appears in the first order correction.