Optimal portfolio and certainty equivalence estimator for the appreciation rate

We study the optimal portfolio selection problem for the class of strategies which do not use direct observations of the appreciation rates of the prices, but rather use historical prices. We consider a multi-stock incomplete diffusion market model with random coefficients. An explicit solution exploring a modification of certainty equivalence principle is found for case of power utilities and for a case when the problem can be embedded to a Markovian setting. Some new estimators for the appreciation rates are given.