Log-optimality with small liability stream

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.