Multivariate utility maximization under transaction costs

We consider a multivariate utility maximization problem in a general multiasset financial market with proportional transaction costs. Unlike the univariate utility case, it is essentially important to avoid the so-called money illusion in the multivariate utility framework. Our utility function depends on the physical amount of the assets rather than the market value of the assets. As such, our utility function can be interpreted as a direct utility function in microeconomics. We generalize the convex duality theory of Kramkov-Schachermayer [Ann. Appl. Probab., 9 (1999), pp. 904-950] to our multivariate utility setting.