Conjugate duality in problems of constrained utility maximization

We show that a simple and elegant method of Bismut [J. Math. Analysis Appl., 44 (1973), pp. 384-404] for applying conjugate duality to convex problems of Bolza adapts directly to problems of utility maximization with portfolio constraints in mathematical finance. This gives a straightforward construction of an associated dual problem together with Euler-Lagrange and transversality relations, which are then used to establish existence of optimal portfolios in terms of solutions of the dual problem. The approach is completely synthetic, and does not require the rather difficult a priori hypothesis of a fictitious complete market for unconstrained optimization, which has been the standard approach for synthesizing optimal portfolios in problems of utility maximization with trading constraints. It also complements a duality synthesis of Rogers [Lecture Notes in Mathematics, No. LNM-1814, Springer-Verlag, New York, 2003, pp. 95-131] and Klein and Rogers [Math. Finance, 17 (2007), pp. 225-247] for general problems of utility maximization with market imperfections.