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Robust utility maximization in non-dominated models with 2BSDEs

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  • Anis Matoussi
  • Dylan Possamai
  • Chao Zhou

Abstract

The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2].

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  • Anis Matoussi & Dylan Possamai & Chao Zhou, 2012. "Robust utility maximization in non-dominated models with 2BSDEs," Papers 1201.0769, arXiv.org, revised Apr 2015.
    • Handle: RePEc:arx:papers:1201.0769
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    References listed on IDEAS

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    1. Lars Peter Hansen & Thomas J Sargent, 2014. "A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection," World Scientific Book Chapters, in: UNCERTAINTY WITHIN ECONOMIC MODELS, chapter 4, pages 83-143, World Scientific Publishing Co. Pte. Ltd..
    2. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
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    Cited by:

    1. Samuel Drapeau & Peng Luo & Dewen Xiong, 2017. "Characterization of Fully Coupled FBSDE in Terms of Portfolio Optimization," Papers 1703.02694, arXiv.org, revised Sep 2019.

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