  Two major financial market complexities are transaction costs and uncertain
volatility, and we analyze their joint impact on the problem of portfolio
optimization. When volatility is constant, the transaction costs optimal
investment problem has a long history, especially in the use of asymptotic
approximations when the cost is small. Under stochastic volatility, but with no
transaction costs, the Merton problem under general utility functions can also
be analyzed with asymptotic methods. Here, we look at the long-run growth rate
problem when both complexities are present, using separation of time scales
approximations. This leads to perturbation analysis of an eigenvalue problem.
We find the first term in the asymptotic expansion in the time scale parameter,
of the optimal long-term growth rate, and of the optimal strategy, for fixed
small transaction costs.
