Mean-Variance Allocation as a Linear Functional on the Signature of the Augmented Path

The paper represents a trading strategy as a linear functional of the signature of an augmented path (prices together with exogenous signals) and derives a closed-form dynamic mean-variance solution in this representation, with drawdown control emerging naturally. We read it as a path-dependent generalization of Markowitz in which factor timing and path memory are encoded in one linear object.

Position = ⟨ℓ, 𝕊(Ẑ)_{0,t}⟩ with Ẑ the lead-lag-augmented path of assets and signals; the mean-variance objective becomes a quadratic form in ℓ whose moments are expected signatures.

The elegance is that path-dependence collapses to linear algebra on expected signatures, so the optimization stays convex while the strategy class is rich. It is the portfolio-construction analogue of the signature pricing results.

Suggests a disciplined way to incorporate path-dependent signals into sizing without abandoning a closed-form, auditable optimizer — consistent with the desk preference for explicit objectives over opaque policies.

Expected-signature moments must be estimated and are sensitive to truncation order and non-stationarity; the mean-variance criterion remains as fragile as its estimated inputs.