Horizon dependence of utility optimizers in incomplete models

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.