  This paper investigates dynamic and static fund separations and their
stability for long-term optimal investments under three model classes. An
investor maximizes the expected utility with constant relative risk aversion
under an incomplete market consisting of a safe asset, several risky assets,
and a single state variable. The state variables in two of the model classes
follow a 3/2 process and an inverse Bessel process, respectively. The other
market model has the partially observed state variable modeled as an
Ornstein-Uhlenbeck state process. We show that the dynamic optimal portfolio of
this utility maximization consists of m+3 portfolios: the safe asset, the
myopic portfolio, the m time-independent portfolios, and the intertemporal
portfolio. Over time, the intertemporal portfolio eventually vanishes, leading
the dynamic portfolio to converge to m+2 portfolios, referred to as the static
portfolio. We also prove that the convergence is stable under model parameter
perturbations. In addition, sensitivities of the intertemporal portfolio with
respect to small parameters perturbations also vanish in the long run. The
convergence rate for the intertemporal portfolio and its sensitivities are
computed explicitly for the presented models.
