This paper studies how to value zero-coupon bonds when the short-term interest rate depends on the bond price itself and when the volatility of the economy is uncertain. The authors frame the problem as a self-consistency condition: bond prices must equal their discounted expected payoffs, but the discount rule depends on the price being computed. To handle volatility ambiguity they use the G-expectation framework, which replaces a single probabilistic model with a family of volatility scenarios and a corresponding sublinear expectation.

To turn the circular valuation into a tractable equation the authors take a logarithm of the bond price. That change leads to a backward stochastic differential equation under G-expectation (a “G‑BSDE”) whose generator has a quadratic term. The quadratic term comes from the log transform, and an extra decreasing G-martingale term records the nonlinear effects of volatility uncertainty. In this way the self-consistent price becomes a solution of a quadratic G‑BSDE on a finite or infinite time horizon.

The mathematical contribution is to show that this nonlinear fixed-point problem is well posed under standard regularity assumptions. For bounded terminal data and Lipschitz conditions on the feedback, they prove existence, uniqueness, comparison, and stability of bounded finite-horizon solutions. With an extra strict monotonicity condition they obtain a unique bounded infinite-horizon solution and show that finite-horizon approximations converge exponentially fast on every compact time interval. These results rely on estimates tailored to quadratic growth and on methods from the recent literature on quadratic G‑BSDEs.

The paper also shows how to use these results for design questions. At a fixed maturity the authors construct discount-rate coefficients that reproduce a prescribed smooth bond-price target. For long maturities they prescribe a target yield and a bounded target profile phi(x) for the compensated logarithmic price (roughly, log price plus time-to-maturity) and then design state-dependent feedback rules so that the compensated log price converges exponentially to phi. Concretely, they obtain bounds of the form |Y_u^T - phi(X_u)| ≤ C e^{-k (T-s)} (quasi‑surely), which give quantitative control of the error as maturity T grows, and ensure the asymptotic yield equals the specified target.