We analyze the martingale selection problem of Rokhlin (2006) in a pointwise (robust) setting. We derive conditions for solvability of this problem and show how it is related to the classical no-arbitrage deliberations. We obtain versions of the Fundamental Theorem of Asset Pricing in examples spanning frictionless markets, models with proportional transaction costs and also models for illiquid markets. In all these examples, we also incorporate trading constraints.
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