Consumption and Investment in Incomplete Markets with Epstein-Zin Preferences in Infinite Horizon

We solve the optimal consumption-investment problem in incomplete markets, where the investor aims to maximise an Epstein-Zin type stochastic differential utility from consumption over an infinite time horizon. We verify that the optimal strategies can be characterised by the limit of a sequence of solutions of the HJB equation in bounded domains with carefully designed boundary conditions. We also conduct numerical experiments for three market models. Explicit solutions are constructed under some parameter regimes.