Convex duality in optimal investment under illiquidity

We study the problem of optimal investment by embedding it in the general conjugate duality framework of convex analysis. This allows for various extensions to classical models of liquid markets. In particular, we obtain a dual representation for the optimum value function in the presence of portfolio constraints and nonlinear trading costs that are encountered e.g. in modern limit order markets. The optimization problem is parameterized by a sequence of financial claims. Such a parameterization is essential in markets without a numeraire asset when pricing swap contracts and other financial products with multiple payout dates. In the special case of perfectly liquid markets or markets with proportional transaction costs, we recover well-known dual expressions in terms of martingale measures.

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