Log-optimality with small liability stream

TL;DR

This paper develops a detailed mathematical expansion of the optimal investment value and wealth process in an incomplete market with non-traded endowments, using advanced duality and projection techniques.

Contribution

It provides a fourth-order expansion of the primal value function and a second-order expansion of the optimal wealth process in the context of incomplete markets with non-traded endowments.

Findings

Derived fourth-order expansion of the primal value function.

Expanded the optimal wealth process up to second order.

Unified treatment for finite and infinite horizons.

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.