Consumption and portfolio optimization with generalized stochastic differential utility in incomplete markets

This paper examines continuous time intertemporal consumption and portfolio choice problems of an investor in a generalized stochastic differential utility preference of Epstein-Zin type with subjective beliefs and with ambiguity, respectively. In these two situations, the aggregators of the utility both contain the "variability"process. Based on the generalized verification theorem which formulates Hamilton-Jacobi-Bellman equations under a weak non-Lipschitz condition, we show the explicit closed-form optimal consumption and portfolio solutions with subjective beliefs and the numerical solutions with ambiguity for Heston model in incomplete market, respectively.