On optimal investment with processes of long or negative memory

TL;DR

This paper investigates utility maximization for investors with power utility in models driven by processes with long or negative memory, providing differentiability results and first-order expansions for non-Markovian models like fractional Brownian motion.

Contribution

It establishes the Frechet differentiability of the utility maximization value with respect to the drift in Banach spaces and derives first-order expansions for non-Markovian processes.

Findings

Frechet differentiability of the value function in drift parameters

First-order expansions for fractional Brownian motion models

Asymptotic results for models with strong mean reversion

Abstract

We consider the problem of utility maximization for investors with power utility functions. Building on the earlier work Larsen et al. (2016), we prove that the value of the problem is a Frechet-differentiable function of the drift of the price process, provided that this drift lies in a suitable Banach space. We then study optimal investment problems with non-Markovian driving processes. In such models there is no hope to get a formula for the achievable maximal utility. Applying results of the first part of the paper we provide first order expansions for certain problems involving fractional Brownian motion either in the drift or in the volatility. We also point out how asymptotic results can be derived for models with strong mean reversion.