  In this paper we find tight sufficient conditions for the continuity of the
value of the utility maximization problem from terminal wealth with respect to
the convergence in distribution of the underlying processes. We also establish
a weak convergence result for the terminal wealths of the optimal portfolios.
Finally, we apply our results to the computation of the minimal expected
shortfall (shortfall risk) in the Heston model by building an appropriate
lattice approximation.
