This paper solves the consumption-investment problem under Epstein-Zin preferences on a random horizon. In an incomplete market, we take the random horizon to be a stopping time adapted to the market filtration, generated by all observable, but not necessarily tradable, state processes. Contrary to prior studies, we do not impose any fixed upper bound for the random horizon, allowing for truly unbounded ones. Focusing on the empirically relevant case where the risk aversion and the elasticity of intertemporal substitution are both larger than one, we characterize optimal consumption and investment strategies through backward stochastic differential equations (BSDEs). Compared with classical results on a fixed horizon, our characterization involves an additional stochastic process to account for the uncertainty of the horizon. As demonstrated in a Markovian setting, this added uncertainty drastically alters optimal strategies from the fixed-horizon case. The main results are obtained through the development of new techniques for BSDEs with superlinear growth on unbounded random horizons.
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