Recursive Utility with Investment Gains and Losses: Existence, Uniqueness, and Convergence

TL;DR

This paper extends recursive utility models to include investment gains and losses, establishing conditions for existence, uniqueness, and convergence of the utility process, and solving related portfolio optimization problems.

Contribution

It introduces a generalized recursive utility model with gain-loss components, proving existence, uniqueness, and convergence results, and extends previous findings to infinite state spaces.

Findings

Unique utility process exists with nonnegative gain-loss utility

Recursive computation converges from any initial guess

Portfolio optimization problem has a unique solution

Abstract

We consider a generalization of the recursive utility model by adding a new component that represents utility of investment gains and losses. We also study the utility process in this generalized model with constant elasticity of intertemporal substitution and relative risk aversion degree, and with infinite time horizon. In a specific, finite-state Markovian setting, we prove that the utility process uniquely exists when the agent derives nonnegative gain-loss utility, and that it can be non-existent or non-unique otherwise. Moreover, we prove that the utility process, when it uniquely exists, can be computed by starting from any initial guess and applying the recursive equation that defines the utility process repeatedly. We then consider a portfolio selection problem with gain-loss utility and solve it by proving that the corresponding dynamic programming equation has a unique solution. Finally, we extend certain previous results to the case in which the state space is infinite.