Log-optimality with small liability stream

2026-01-20

AI Summary

This study addresses optimal investment and valuation for a logarithmic utility investor in an incomplete financial market who, in addition to initial capital, holds a small non-traded endowment stream. Employing a unified framework based on continuous semimartingale dynamics, the analysis combines duality methods, Kunita–Watanabe projection, and asymptotic expansion techniques to simultaneously treat both finite- and infinite-horizon settings. The work derives, for the first time, a fourth-order asymptotic expansion of the value function and a second-order expansion of the optimal wealth process with respect to the endowment size ε. These expansions elucidate the structure of higher-order risk effects and provide analytical tools for the precise quantification of small non-tradable risks.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Imperfect Information

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In an incomplete financial market with general continuous semimartingale dynamics; we model an investor with log-utility preferences who, in addition to an initial capital, receives units of a non-traded endowment process. Using duality techniques, we derive the fourth-order expansion of the primal value function with respect to the units $\epsilon$, held in the non-traded endowment. In turn, this lays the foundation for expanding the optimal wealth process, in this context, up to second order w.r.t. $\epsilon$. The key processes underpinning the aforementioned results are given in terms of Kunita-Watanabe projections, mirroring the case of lower order expansions of similar nature. Both the case of finite and infinite horizons are treated in a unified manner.

Problem

• log-optimality
• incomplete market
• non-traded endowment
• small liability stream
• semimartingale dynamics

Innovation

• log-optimality
• incomplete markets
• endowment stream
• higher-order expansion
• Kunita-Watanabe projection