Linear independence properties of the signature components of time-augmented stochastic processes

Adding the time as a component of a stochastic process before computing its signature terminal value ensures injectivity and supports universal approximation results, but it induces linear dependence among the components of the signature terminal value. For any natural number $N$, the terminal values of the signature components associated with words of length not greater than $N$ are the image of the terminal values of the signature components associated with words of length $N$ by some universal linear map. We generalize this result by exhibiting other subfamilies of components -- represented by subfamilies of words -- with the same representation property. When considering the signature of the solution to a stochastic differential equation with additive Brownian noise, we show that any such subfamily of components is linearly independent for the almost-sure equality and therefore provides a basis of the linear span of all components associated with words of length not greater than $N$. The linear independence of these subfamilies is preserved for the affine interpolation of this solution on a grid with a sufficiently small time step. We characterize bases of components with minimal computation cost. Finally, we remark that the subfamilies of words obtained above share a similar representation property when applied to the time-augmented EFM signature. For a Brownian semimartingale with a non-degenerate diffusion coefficient, we show that any such subfamily of components of its time-augmented EFM signature is almost-surely linearly independent for the $dt$-a.e. equality.