Existence of shadow prices in finite probability spaces

A shadow price is a process $${\widetilde{S}}$$ lying within the bid/ask prices $${\underline{S},\overline{S}}$$ of a market with proportional transaction costs, such that maximizing expected utility from consumption in the frictionless market with price process $${\widetilde{S}}$$ leads to the same maximal utility as in the original market with transaction costs. For finite probability spaces, this note provides an elementary proof for the existence of such a shadow price.

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