  This paper studies convex duality in optimal investment and contingent claim
valuation in markets where traded assets may be subject to nonlinear trading
costs and portfolio constraints. Under fairly general conditions, the dual
expressions decompose into tree terms, corresponding to the agent's risk
preferences, trading costs and portfolio constraints, respectively. The dual
representations are shown to be valid when the market model satisfies an
appropriate generalization of the no-arbitrage condition and the agent's
utility function satisfies an appropriate generalization of asymptotic
elasticity conditions. When applied to classical liquid market models or models
with bid-ask spreads, we recover well-known pricing formulas in terms of
martingale measures and consistent price systems. Building on the general
theory of convex stochastic optimization, we also derive optimality conditions
in terms of an extended notion of a "shadow price".
