Log-Optimality Achieves Second-Order Wealth Expansion with Fourth-Order Endowment Precision

Scientists are tackling the challenge of optimal investment strategies in incomplete financial markets, where traditional approaches often fail. Michail Anthropelos, Constantinos Kardaras, and Constantinos Stefanakis, from the Hellenic Foundation for Research and Innovation, present a novel framework based on log-utility preferences and a continuous semimartingale model to analyse portfolios with a non-traded endowment. Their research derives a fourth-order expansion of the primal value function and a second-order expansion of the optimal wealth process, providing unprecedented precision in modelling wealth dynamics. This framework applies uniformly to both finite and infinite investment horizons, offering a unified solution grounded in Kunita-Watanabe projections.

The study introduces a new method to examine log-optimality in the presence of small liability streams, a common scenario in real-world finance. Using duality techniques, the team expands the primal value function with respect to the endowment units, ε, enabling the optimal wealth process to be approximated up to second order. This extension provides a finer-grained understanding of optimal investment behaviour and demonstrates that the first-order approximation remains optimal in complete markets, offering a benchmark for analysing market incompleteness.

By employing Kunita-Watanabe projections, the researchers maintain consistency with lower-order expansions while allowing for arbitrarily large time horizons. This “long-horizon asymptotic” approach eliminates dependence on specific maturities, making the results particularly relevant for pension funds and long-term liabilities. Experiments confirm that the expansions are robust for both finite and infinite horizons, providing a comprehensive analytical tool for modelling investor behaviour in incomplete markets with non-traded endowments.

In essence, this work delivers a powerful, unified framework for understanding log-optimal wealth allocation, extending prior results [KS06, KS07] to log-utility preferences and offering valuable insights into asset allocation under market imperfections. The methodology paves the way for more precise long-term investment strategies, bridging theoretical advances with practical applications in financial risk management.

Endowment Wealth Expansion via Duality Techniques offers significant

Scientists have developed a fourth-order expansion of the primal value function for an investor in an incomplete financial market with a non-traded endowment process. This research expands the optimal wealth process up to second order with respect to the units held in the endowment, utilising Kunita-Watanabe projections, a technique mirroring expansions in similar contexts. The work addresses both finite and infinite investment horizons within a unified framework, offering a robust analytical approach. Researchers successfully applied duality techniques to analyse an investor with log-utility preferences and a non-traded endowment, building upon established financial economics theory dating back to Merton’s initial studies of optimal investment.

This advancement tackles the complexities of incomplete markets where perfect replication of assets is impossible due to real-world frictions like transaction costs and non-traded assets, a challenge previously addressed by Cvitanić, Schachermayer, and Wang, among others. The authors acknowledge that their analysis relies on specific mathematical conditions and the modelling of the endowment process, potentially limiting the direct applicability to all real-world scenarios. Future research could explore the implications of relaxing these assumptions or extending the model to incorporate more complex market dynamics and investor preferences.