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In this article we study a multi-asset version of the Merton investment andconsumption problem with proportional transaction costs. In general it isdifficult to make analytical progress towards a solution in such problems, butwe specialise to a case where transaction costs are zero except for sales andpurchases of a single asset which we call the illiquid asset.  Assuming agents have CRRA utilities and asset prices follow exponentialBrownian motions we show that the underlying HJB equation can be transformedinto a boundary value problem for a first order differential equation. Theoptimal strategy is to trade the illiquid asset only when the fraction of thetotal portfolio value invested in this asset falls outside a fixed interval.Important properties of the multi-asset problem (including when the problem iswell-posed, ill-posed, or well-posed only for large transaction costs) can beinferred from the behaviours of a quadratic function of a single variable andanother algebraic function.

A multi-asset investment and consumption problem with transaction costs