Multivariate utility maximization with proportional transaction costs and random endowment

In this paper we deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to modeling a currency market with proportional transaction costs). In particular, we extend the results in [L. Campi and M. Owen, Finance Stoch., 15 (2011), pp. 461-499] to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. We start by studying some basic properties of the value function (which is now defined on a space of random variables), and then we dualize the problem following some convex analysis techniques which have proven very useful in this field of research. We finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. The last section of the paper is devoted to an application of our results to utility indifference pricing.