Admissibility (convergence) of the CIR signature functional

Prove that the element b_CIR defined implicitly by the non-linear algebraic relation (b_CIR) {2} = v · 1 + ((κθ − η^2/4) · 1 − κ (b_CIR) {2}) {1} + η b_CIR {2} belongs to the admissible set A of extended tensor algebra elements for which <·, Sig_t> is absolutely convergent for all t ∈ [0,T]. Establishing b_CIR ∈ A would provide the theoretical convergence needed to justify the linear-signature representation of the CIR process.

Background

In the signature framework, infinite linear combinations are well-defined only for coefficients lying in the admissible set A, which ensures absolute convergence of the series defining <a, Sig_t>. For OU and mean-reverting GBM models, the required admissibility has been established, enabling exact representations.

For the CIR case, the representation involves an element defined via a non-linear identity. Demonstrating that this element lies in A is necessary to justify the representation rigorously. Although the authors provide numerical evidence, a proof of admissibility remains outstanding.