Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case.

*(English)*Zbl 0736.90017Summary: We employ a martingale approach to study a dynamic consumption-portfolio problem in continuous time with incomplete markets and short-sale constraints. We introduce a notion of minimax local martingale and transform the dynamic problem into a static problem of maximizing expected utility over the consumption bundles that satisfy a single budget constraint formed using that measure. We establish the existence of and characterize the minimax local measure, provide sufficient conditions for the dynamic consumption-portfolio problem to have a solution, and relate the optimal policies to the solution of quasi-linear partial differential equation.

##### MSC:

91B62 | Economic growth models |

91B28 | Finance etc. (MSC2000) |

##### Keywords:

martingale approach; dynamic consumption-portfolio problem; continuous time; incomplete markets; short-sale constraints; minimax local martingale; quasi-linear partial differential equation
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\textit{H. He} and \textit{N. D. Pearson}, J. Econ. Theory 54, No. 2, 259--304 (1991; Zbl 0736.90017)

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