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Mathematical Problems in Engineering
Volume 2013, Article ID 359701, 8 pages
Research Article

State-Dependent Utilities and Incomplete Markets

Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia

Received 23 January 2013; Accepted 1 May 2013

Academic Editor: Guangchen Wang

Copyright © 2013 Jaime A. Londoño. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The problem of optimal consumption and investment for an agent that does not influence the market is solved. The optimization criteria are based on a state-dependent utility functional as proposed in Londoño (2009). The proposed solution is given in any market without state-tame arbitrage opportunities, includes several utilities structures, and includes incomplete markets where there are multiple state variables. The solutions obtained for optimal wealths consumptions, and portfolios are explicit and easily computable; the main condition for the result to hold is that the income process of each agent is hedgeable, requiring a natural condition on employer and employee to agree on a contract whose risk can be managed by both parties. In this paper we also developed a theory of markets when the processes are generalization of Brownian flows on manifolds, since this framework shows to be the natural one whenever the problem of intertemporal equilibrium is addressed.