We consider a problem of optimal investment with intermediate consumption in the framework of an incomplete semimartingale model of a financial market. We show that a necessary and sufficient condition for the validity of key assertions of the theory is that the value functions of the primal and dual problems are finite.
Thesis Advisor: Professor Dmitry Kramkov
AcknowledgementsFirst and foremost, I would like to thank my advisor Dmitry Kramkov. He has systematically challenged me to consider the most general case in which one still can hope to achieve a result. Dmitry has helped me to overcome difficulties with his advice and provided critical insights and suggestions at the right moments. Studying and discussing with him, in particular his papers, gave me an idea of how central mathematical results are developed and helped to perform on my (local) maximum.I would also like to thank Steven Shreve for teaching so much through his books and our conversations. Many other professors in the department have given me invaluable instruction and advice, including
We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk (NUPBR) and of the finiteness of both primal and dual value functions.2010 Mathematics Subject Classification. 91G10, 93E20. JEL Classification: C60, G1. Key words and phrases. Utility maximization, arbitrage of the first kind, local martingale deflator, duality theory, semimartingale, incomplete market.
In the framework of an incomplete financial market where the stock price dynamics are modeled by a continuous semimartingale (not necessarily Markovian) an explicit second-order expansion formula for the power investor's value function -seen as a function of the underlying market price of risk process -is provided. This allows us to provide first-order approximations of the optimal primal and dual controls. Two specific calibrated numerical examples illustrating the accuracy of the method are also given.2010 Mathematics Subject Classification. Primary 91G10, 91G80; Secondary 60K35. Journal of Economic Literature (JEL) Classification: C61, G11.
We consider an optimal investment problem with intermediate consumption and random endowment, in an incomplete semimartingale model of the financial market. We establish the key assertions of the utility maximization theory, assuming that both primal and dual value functions are finite in the interiors of their domains and that the random endowment at maturity can be dominated by the terminal value of a self-financing wealth process. In order to facilitate the verification of these conditions, we present alternative, but equivalent conditions, under which the conclusions of the theory hold.
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