Horizon dependence of utility optimizers in incomplete models

TL;DR

This paper investigates how utility maximization in incomplete models depends on the time horizon, revealing continuity properties and conditions affecting the convergence of optimal strategies as horizons change.

Contribution

It establishes the continuity of the value function and optimal wealth with respect to the time horizon, and identifies conditions where expected utility convergence fails.

Findings

Primal value function is continuous in the time horizon.

Optimal terminal wealth varies continuously with horizon changes.

Expected utility may not converge when applying shorter horizon optimizers to longer horizons.

Abstract

This paper studies the utility maximization problem with changing time horizons in the incomplete Brownian setting. We first show that the primal value function and the optimal terminal wealth are continuous with respect to the time horizon $T$. Secondly, we exemplify that the expected utility stemming from applying the $T$-horizon optimizer on a shorter time horizon $S$, $S < T$, may not converge as $S\uparrow T$ to the $T$-horizon value. Finally, we provide necessary and sufficient conditions preventing the existence of this phenomenon.