  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.
