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arXiv · 1204.0305

Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs

Jin Hyuk Choi·Mihai Sirbu·Gordan Zitkovic
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Abstract

We revisit the optimal investment and consumption model of Davis and Norman (1990) and Shreve and Soner (1994), following a shadow-price approach similar to that of Kallsen and Muhle-Karbe (2010). Making use of the completeness of the model without transaction costs, we reformulate and reduce the Hamilton-Jacobi-Bellman equation for this singular stochastic control problem to a non-standard free-boundary problem for a first-order ODE with an integral constraint. Having shown that the free boundary problem has a smooth solution, we use it to construct the solution of the original optimal investment/consumption problem in a self-contained manner and without any recourse to the dynamic programming principle. Furthermore, we provide an explicit characterization of model parameters for which the value function is finite.

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Jin Hyuk Choi, Mihai Sirbu, Gordan Zitkovic. 2012-04-02. Shadow prices and well-posedness in the problem of optimal investment and consumption with transaction costs. https://arxiv.org/abs/1204.0305

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