  We present an optimal investment theorem for a currency exchange model with
random and possibly discontinuous proportional transaction costs. The
investor's preferences are represented by a multivariate utility function,
allowing for simultaneous consumption of any prescribed selection of the
currencies at a given terminal date. We prove the existence of an optimal
portfolio process under the assumption of asymptotic satiability of the value
function. Sufficient conditions for asymptotic satiability of the value
function include reasonable asymptotic elasticity of the utility function, or a
growth condition on its dual function. We show that the portfolio optimization
problem can be reformulated in terms of maximization of a terminal liquidation
utility function, and that both problems have a common optimizer.
