Expected signatures via partial integration, coordinate change and symmetrization

Abstract

We study signature transformations of heterogeneous paths $Y=(A,X)$ whose components may differ in regularity and probabilistic structure. We introduce an invertible change of coordinates $Ψ$ such that the transformed signature $Ψ\circ\mathrm{Sig}$ eliminates mixed integration against the irregular component $X$ and admits a representation in terms of signature coordinates of $X$ and iterated integration against the regular component $A$. In addition, we exploit this representation to further represent partially symmetrized signatures. Our main application concerns new expected signature formulas for processes with deterministic augmentation. On the analytical side, these formulas are leveraged to study moment problems. On the numerical side, they enable accurate computation of expected signatures, thereby overcoming typical computational bottlenecks in applications. We illustrate these advantages in a signature-based stochastic control problem driven by fractional Brownian motion.