This paper studies convex duality in optimal investment and contingent claimvaluation in markets where traded assets may be subject to nonlinear tradingcosts and portfolio constraints. Under fairly general conditions, the dualexpressions decompose into tree terms, corresponding to the agent's riskpreferences, trading costs and portfolio constraints, respectively. The dualrepresentations are shown to be valid when the market model satisfies anappropriate generalization of the no-arbitrage condition and the agent'sutility function satisfies an appropriate generalization of asymptoticelasticity conditions. When applied to classical liquid market models or modelswith bid-ask spreads, we recover well-known pricing formulas in terms ofmartingale measures and consistent price systems. Building on the generaltheory of convex stochastic optimization, we also derive optimality conditionsin terms of an extended notion of a "shadow price".
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