Optimal consumption from investment and random endowment in incomplete semimartingale markets.

*(English)*Zbl 1076.91017The authors consider the utility maximization problem in continuous-time financial markets. More exactly, their market is a constrained incomplete semimartingale one with a random endowment process. The main idea is, as usual, to use the convex nature of the problem, to solve a dual variational problem and then proceed as in the complete case. First, the intertemporal consumption is incorporated in the optimization problem. Then, the agent is considered who invests in an incomplete market, where prices are modelled by an arbitrary semimartingale with right-continuous and left-limited paths. For utility functions the concept of asymptotic elasticity is introduced and under appropriate condition of “reasonable asymptotic elasticity”, the existence and uniqueness of optimal consumption-investment strategies is established. To make the duality approach possible, a detailed characterization of the enlarged dual domain is provided. As application, two special cases are treated: a constrained Itô process market where it is proved that the optimal dual process is always a local martingale, and a “totally incomplete” market where the agent is not allowed to invest in the stock market at all.

Reviewer: Yuliya S. Mishura (Kyïv)

##### MSC:

91G10 | Portfolio theory |

60G07 | General theory of stochastic processes |

60G44 | Martingales with continuous parameter |

91B70 | Stochastic models in economics |

##### Keywords:

utility maximization; random endowment; incomplete markets; convex duality; stochastic processes; finitely additive measures
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\textit{I. Karatzas} and \textit{G. Žitković}, Ann. Probab. 31, No. 4, 1821--1858 (2003; Zbl 1076.91017)

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