Power-Utility Investment with Long-Memory and Mean-Reverting Processes

Summary

This paper studies utility maximization for investors with power utility when price dynamics have long-memory or non-Markovian features. It proves that the value of the investment problem changes in a Fréchet-differentiable way with the drift, under a condition that the drift belongs to a suitable Banach space. This result provides a foundation for approximating utility outcomes when closed-form solutions are unavailable.

The authors apply the result to models involving fractional Brownian motion in either the drift or volatility, deriving first-order expansions. They also describe how asymptotic results can be obtained for models with strong mean reversion. The summary does not give the expansions themselves, parameter restrictions, or numerical illustrations, and it notes that maximal utility generally has no explicit formula in the non-Markovian setting. The contribution is therefore analytical rather than a tested investment strategy.

Key ideas

• The study considers utility maximization for investors with power utility.
• The value function is proved Fréchet-differentiable with respect to drift under a suitable function-space condition.
• First-order approximations are developed for models with fractional Brownian motion in drift or volatility.
• Asymptotic analysis is also discussed for models with strong mean reversion.

Tags

# On optimal investment with processes of long or negative memory


# On optimal investment with processes of long or negative memory









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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.