Conjectured signature-based linear representation of the CIR process

Establish that the one-dimensional Cox–Ingersoll–Ross (CIR) process V defined by dV_t = κ(θ − V_t) dt + η √V_t dW_t with V_0 = v > 0 admits the signature-based linear representation V_t = <c_CIR, Sig_t> = (<b_CIR, Sig_t>)^2, where Sig_t is the signature of the time-augmented Brownian motion (t, W_t), c_CIR := (b_CIR) {2}, and b_CIR is an element of the extended tensor algebra that satisfies the non-linear algebraic identity (b_CIR) {2} = v · 1 + ((κθ − η^2/4) · 1 − κ (b_CIR) {2}) {1} + η b_CIR {2}.

Background

The paper develops signature volatility models in which processes are expressed as possibly infinite linear combinations of elements of the signature of the time-extended Brownian motion. Several classical models (e.g., Ornstein–Uhlenbeck and mean-reverting geometric Brownian motion) are shown to admit exact linear representations in this framework.

For the Cox–Ingersoll–Ross (CIR) process, the authors propose a specific form of representation in which V_t is the square of a signature-based linear functional. The proposed representation is specified via a non-linear algebraic relation for an element of the extended tensor algebra, but a rigorous proof that this construction indeed reproduces the CIR dynamics has not been provided.