Kim and Park prove that bond-price feedback produces a unique bounded pricing fixed point for one maturity. A desk still lacks the estimated rate rule and uncertainty set needed to use it. Their estimate constants remain unevaluated, described only through C_1, C_2, sigma_lower, sigma_upper, T and the reverse-Hölder order q. The paper separately leaves numerical approximation of the pricing fixed point for future work.

The price sets its own discount rate

Ordinarily, a zero-coupon bond discounts its terminal payoff using a short rate fixed before valuation. Kim and Park let that rate depend on time, a state process X and the bond's own log price, Y_s = log P(s,T). They suggest macro factors such as inflation and unemployment for X. Price now enters the rule that determines price. As the paper says, "Unlike the standard bond-pricing formula, (5.2) is implicit: the short rate depends on the bond price through Y."

Discounting also runs through two channels. The coefficient r accrues against ordinary time. A family rho_ij accrues against realized quadratic covariation d<B^i,B^j>. Under G-expectation, strong ellipticity confines quadratic variation between sigma_lower^2 and sigma_upper^2, leaving those channels attached to different objects. A single-measure model pins down quadratic variation, so the split collapses.

The paper treats rho_ij specifically as a coefficient in the discounting rule. Hölzermann's (2022) drift condition uses market prices of uncertainty, which are separate objects. Kim and Park impose that condition by assumption through a lambda under which discounted bonds are symmetric G-martingales.

Taking logs turns the fixed-point problem into a quadratic G-BSDE with terminal condition zero. Its generator places (1/2)Z^i Z^j minus rho_ij against d<B^i,B^j>. A decreasing G-martingale K carries the volatility-uncertainty component. The rest of the paper studies this equation.

What the theorems deliver

Finite maturity requires Assumptions 3.1 and 5.1. They make r and rho_ij Lipschitz in (x,y) with constant C_1 and bounded at y = 0 by C_2. These conditions give a unique bounded solution, hence a unique self-consistent log price.

The proof draws on the standard quadratic-BSDE machinery under sublinear expectation. Applying Itô's formula to e^{-gamma Y}, with gamma = 6C_1, bounds Z in G-BMO. The estimate E[(integral |Z|^2 d<B>)^n] <= n! ||Z||_BMO^n then places K in every L^p, after which a comparison theorem follows. The value function u(t,x) = Y_t^{t,x} is Lipschitz in x. Its time regularity satisfies |u(t,x) - u(t+delta,x)| <= C(1+|x|) delta^{1/2}.

Infinite maturity adds a strict monotonicity constant mu > 0 in the backward variable. The finite-horizon approximations then obey |Y^n_s| <= C_2/mu, and their bounded infinite-horizon limit keeps the same bound. Convergence is quasi-sure, with sup|Y^T - Y| <= (C_2/mu) e^{-mu(T-s)}. Here T - s means remaining maturity. The authors explicitly caution against interpreting mu as the adjustment speed after a policy move.

Design comes before implementation

Policy readers will likely begin with the inverse problem. For a fixed maturity, choose a target psi in C^{1,2} satisfying psi(T,.) = 0. Matching its G-Itô dynamics then determines r and rho_ij from the derivatives of psi.

Long maturities use a prescribed yield lambda > 0 and a bounded profile phi in C^2. Adding the restoring term mu(y - phi(x)) to the rate makes the infinite-horizon solution exactly Y = phi(X). Consequently, e^{lambda(T-s)} P(s,T) tends to e^{phi(X_s)}, while the asymptotic yield equals lambda.

The claim stays deliberately limited. The authors describe lambda as a chosen design target rather than an ergodic eigenvalue produced by the model. A modeler supplies lambda, phi and mu. The theorem provides a rate rule that reproduces those choices.

Constants without usable values

The first obstacle is quantitative. Proposition 3.5 gives a constant that depends on the reverse-Hölder order q. Remark 3.1 adds that q depends on ||Z||_BMO, itself an output of the equation. Lemma 2.2 does provide an explicit threshold function, (1 + (1/q^2) log((2q-1)/(2(q-1))))^{1/2} - 1, though the associated C_q remains unspecified. Theorem 3.7 states dependence on C_1, C_2, sigma_lower, sigma_upper and T, omitting q from that list. A designer directly chooses only C_2/mu and mu.

Existence proceeds by compactness after two truncations. First g_n = min(g, n), then g_{n,l} = max(g_n, -l), followed by the Lipschitz cutoff iota_k. The resulting generator g_{n,l,k} is Lipschitz in z with constant C_1(1+4k) + 2(n+l)/k. Its solution satisfies |Y^{n,l,k}| <= C_2 + (n+l) sigma_upper T. Since that bound grows with n+l, the limiting argument passes through Arzelà-Ascoli on the value functions. Theorem 3.11 characterizes u(t,x) as the viscosity solution of a fully nonlinear PDE, without supplying a solution method.

Boundedness carries real restrictions. Uniqueness applies within bounded Y. In the long-maturity construction, r(s,x,0) = <Dphi(x), b(s,x)> - mu phi(x) + lambda must stay bounded by C_2 and remain Lipschitz in x. For a mean-reverting state, those requirements exclude an affine profile phi and force its gradient to decay.

The sign restriction also matters. Every result uses mu > 0, meaning the rule raises the policy rate when the compensated log price stands above its target. We did not find a treatment of the accommodative sign. Most displayed proofs set d = 1, m = 1, f = 0 "for simplicity". In precisely that case, the rho_ij cross terms that distinguish the discounting rule disappear.

And the construction covers one reference maturity.

The curve remains outside the theorem

The authors acknowledge the missing cross-maturity condition. Simultaneously traded maturities must remain compatible with a common money-market account, and that requirement is unresolved here. A rule responding to the ten-year price also determines prices elsewhere on the curve.

They make the associated arbitrage limitation equally clear: "Uniqueness of its pricing fixed point does not by itself establish a unique equilibrium price or the symmetric-martingale conditions needed to rule out arbitrage in a traded bond market." Their response is to present the paper as groundwork for later extensions. Within one backward-equation framework, it connects valuation under volatility uncertainty, horizon stability and target-oriented short-rate design.

We could not test any of it. We have no cash zero-coupon series. More decisively, the paper supplies no estimated functional form for r or rho_ij and no uncertainty set. Gamma, equivalently sigma_lower and sigma_upper, enters exogenously, while its statistical specification appears among the authors' future-work items. Treasury futures would substitute a deliverable basket and implied financing, a different object from this self-consistent zero-coupon discounting loop.

The gap matches the one we identified in a rough-Heston microfoundation whose parameters prices could not invert (our note). The mechanism is coherent, yet observable prices provide no route back to it. One worked scalar example would help. Choose an Ornstein-Uhlenbeck state, a sigma band, a mu and a phi. Then show the finite-maturity solution approaching phi(X) at e^{-mu(T-s)} as maturity increases. Such a plot would mark the distance between a well posed fixed-point problem and a model a desk could debate.