In this article, we consider the problem of robust utility maximization in an incomplete market with volatility uncertainty  . The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second order backward stochastic differential equations introduced in Soner, Touzi and Zhang (2010) for Lipschitz continuous generator, then generalized by Possamai and Zhou (2011) in the quadratic growth case. We solve the problem for exponential, power and logarithmic utility functions and prove existence of an optimal strategy. Finally we provide several examples which shed more light on the problem and its links with the classical utility maximization one   .
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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In this article, we consider the
The
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is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds
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to study
studying
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second order
second-order
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introduced in Soner, Touzi and Zhang (2010) for Lipschitz continuous generator, then generalized by Possamai and Zhou (2011) in the quadratic growth case. We solve the problem
(2BSDEs for short) with quadratic growth generators. We show
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utility functions
utilities that the value function of the problem can be written as the initial value of a particular 2BSDE
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we provide
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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
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