Epstein-Zin Utility Maximization on a Random Horizon

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

Robust maximization of asymptotic growth under covariance uncertainty

This paper resolves a question proposed in Kardaras and Robertson [Ann. Appl. Probab. 22 (2012) 1576-1610]: how to invest in a robust growth-optimal way in a market where precise knowledge of the covariance structure of the underlying assets is unavailable. Among an appropriate class of admissible covariance structures, we characterize the optimal trading strategy in terms of a generalized version of the principal eigenvalue of a fully nonlinear elliptic operator and its associated eigenfunction, by slightly restricting the collection of nondominated probability measures.Comment: Published in at http://dx.doi.org/10.1214/12-AAP887 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

General Stopping Behaviors of Naive and Non-Committed Sophisticated Agents, with Application to Probability Distortion

We consider the problem of stopping a diffusion process with a payoff functional that renders the problem time-inconsistent. We study stopping decisions of naive agents who reoptimize continuously in time, as well as equilibrium strategies of sophisticated agents who anticipate but lack control over their future selves' behaviors. When the state process is one dimensional and the payoff functional satisfies some regularity conditions, we prove that any equilibrium can be obtained as a fixed point of an operator. This operator represents strategic reasoning that takes the future selves' behaviors into account. We then apply the general results to the case when the agents distort probability and the diffusion process is a geometric Brownian motion. The problem is inherently time-inconsistent as the level of distortion of a same event changes over time. We show how the strategic reasoning may turn a naive agent into a sophisticated one. Moreover, we derive stopping strategies of the two types of agent for various parameter specifications of the problem, illustrating rich behaviors beyond the extreme ones such as "never-stopping" or "never-starting"