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 .
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 the optimal consumption and investment strategies using backward stochastic differential equations with superlinear growth on unbounded random horizons. This characterization, compared with the classical fixed-horizon result, involves an additional stochastic process that serves to capture the randomness of the horizon. As demonstrated in two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies .
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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
two concrete examples, changing from a fixed horizon to a random one drastically alters the optimal strategies