  This paper introduces a dual problem to study a continuous-time consumption
and investment problem with incomplete markets and stochastic differential
utility. For Epstein-Zin utility, duality between the primal and dual problems
is established. Consequently the optimal strategy of the consumption and
investment problem is identified without assuming several technical conditions
on market model, utility specification, and agent's admissible strategy.
Meanwhile the minimizer of the dual problem is identified as the utility
gradient of the primal value and is economically interpreted as the "least
favorable" completion of the market.
