Extend well-definedness and convergence of signature-based state representation beyond Brownian drivers

Establish, for a general driving signal Y (for example, fractional Brownian motion), that the process ⟨ℓ̂, Σ^Y⟩ is well-defined and that the truncated pairings ⟨(ℓ̂)^≤L, Σ^Y⟩ converge to the controlled state process X^u in an appropriate sense, thereby enabling an analogue of the Brownian-motion results for non-Brownian drivers.

Background

The paper proves, for Brownian motion, that signature-parameterized controls induce state processes and costs that can be represented via linear functionals on expected signatures, and that truncated problems converge in value. Extending these results to non-Brownian drivers, such as fractional Brownian motion, is a key motivation, but relies on properties (e.g., growth and convergence of signature expansions) that the authors only establish in the Brownian case.

In the non-Brownian setting, the authors explicitly identify uncertainty about the well-definedness of the process ⟨ℓ̂, Σ⟩ and the convergence of truncated pairings to the controlled state Xu, noting that their Brownian results are essential to the current proofs.

References

It is now unclear whether the process ⟨ℓ̂, Σ⟩ is even well-defined; we need the truncated pairing ⟨(ℓ̂)≤L, Σ⟩ to converge to the process Xu in some sense, and the latter process also needs to have a proper meaning. But our result (Theorem 1) to this effect makes essential use of the Brownian motion and its signature.

— Solving Linear-Quadratic Stochastic Control Problems with Signatures  (2602.23473 - Aqsha et al., 26 Feb 2026) in Subsection "Beyond Brownian noise"