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Some Functional Analytic Tools for Utility Maximization

In: Modern Stochastics and Applications

Author

Listed:
  • Alexander A. Gushchin

    (Steklov Mathematical Institute
    National Research University Higher School of Economics)

  • Ruslan V. Khasanov

    (Lomonosov Moscow State University)

  • Ivan S. Morozov

    (Lomonosov Moscow State University)

Abstract

The aim of the chapter is to extend the application of convex duality methods to the problem of maximizing expected utility from terminal wealth. More precisely, we restrict attention to a dual characterization of the value function of this problem and to a static setting. A general scheme to solve this problem is proposed. In the case where the utility function is finite on $$\mathbb{R}$$ , we use the approach, suggested by Biagini and Frittelli, based on using an Orlicz space constructed from an investor’s utility function. We reduce the original problem to an optimization problem in this space in a nontrivial way, which allows us to weaken essentially assumptions on the model. We also study the problem of utility maximization with random endowment considered by Cvitanić, Schachermayer, and Wang. Using the space ψ L ∞ with a weight function ψ constructed from a random endowment permits us to consider unbounded random endowments. Another important contribution is that in both problems under consideration, we provide versions of the dual problem that are free of singular functionals.

Suggested Citation

  • Alexander A. Gushchin & Ruslan V. Khasanov & Ivan S. Morozov, 2014. "Some Functional Analytic Tools for Utility Maximization," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 267-285, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_15
    DOI: 10.1007/978-3-319-03512-3_15
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