Represent LQ costs using robust signatures by constructing an appropriate tensor functional

Construct a tensor functional J(ℓ) compatible with robust signatures λ(Σ) such that ⟨J(ℓ), λ(Σ)⟩ equals the linear–quadratic cost J(u) when controls are parameterized by u=⟨ℓ, E[λ(Σ)]⟩, thereby enabling the signature-based linear–quadratic optimization framework with robust signatures beyond Brownian motion.

Background

Beyond Brownian motion, the authors note that L^p-universality results for linear functionals of classical signatures are limited. While robust signatures admit such universality, the cost representation used in the paper hinges on pairing a tensor J(ℓ) with the expected signature, and the authors do not know how to construct an analogous J(ℓ) for robust signatures.

This construction would extend the deterministic polynomial optimization representation of linear–quadratic costs to settings where robust signatures are more suitable than classical signatures.