  This paper concerns the recursive utility maximization problem. We assume
that the coefficients of the wealth equation and the recursive utility are
concave. Then some interesting and important cases with nonlinear and nonsmooth
coefficients satisfy our assumption. After given an equivalent backward
formulation of our problem, we employ the Fenchel-Legendre transform and derive
the corresponding variational formulation. By the convex duality method, the
primal "sup-inf" problem is translated to a dual minimization problem and the
saddle point of our problem is derived. Finally, we obtain the optimal terminal
wealth. To illustrate our results, three cases for investors with ambiguity
aversion are explicitly worked out under some special assumptions.
