Epstein-Zin Utility Maximization on a Random Horizon

Abstract

This paper solves the consumption-investment problem with Epstein-Zin utility 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 developing new techniques for BSDEs with superlinear growth on an unbounded random horizon.

Asymptotically Optimal Management of Heterogeneous Collectivised Investment Funds

“…The existence and uniqueness of solutions of the BSDE (1.2) may be established using the technique of the proof of Proposition 2.2 of [18] combined with the properties of BSDEs with default jumps established in [12]. Since we assume that τ is bounded above by T , the techniques of [5] are not required.…”

Optimal consumption and portfolio selection with Epstein-Zin utility under general constraints

“…For time-additive utilities, on the one hand, one can refer to Cvitanic and Karatzas (1992), Rouge and El Karoui (2000) and Bian, Chen and Xu (2019) for convex constraints, and Hu, Imkeller and Müller (2005), Heunis (2015) and Cheridito and Hu (2011) for closed constraints. On the other hand, for recursive utilities, in a market with stochastic investment opportunities, the analysis was developed by El Karoui, Peng and Quenez (2001), Wang, Wang and Yang (2016), Aurand and Huang (2020), Schroder and Skiadas (2003), Schroder and Skiadas (2005), Matoussi, Mezghani and Mnif (2015), Yang, Liang and Zhou (2019) and Melnyk, Muhle-Karbe and Seifried (2020). These papers include a wide variety of constraints such as stochastic income, random horizons, convex constraints, borrowing costs, transaction costs, etc.…”