  We consider the problem of utility maximization for investors with power
utility functions. Building on the earlier work Larsen et al. (2016), we prove
that the value of the problem is a Frechet-differentiable function of the drift
of the price process, provided that this drift lies in a suitable Banach space.
  We then study optimal investment problems with non-Markovian driving
processes. In such models there is no hope to get a formula for the achievable
maximal utility. Applying results of the first part of the paper we provide
first order expansions for certain problems involving fractional Brownian
motion either in the drift or in the volatility. We also point out how
asymptotic results can be derived for models with strong mean reversion.
