Using the elements from the theory of quadratic backward stochastic differential equations with unbounded terminal data, we solve an exponential utility maximization problem with unbounded random endowments under portfolio constraints. Our results thus generalize the previous results of Hu et al. (2005) [Ann. Appl. Probab., 15, 1691--1712] from bounded to unbounded case. As an application, we study utility indifference valuation of financial derivatives with unbounded payoffs, obtaining a new convex dual representation of the prices and their asymptotics for the risk aversion parameter.
↧