  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].
