Discount-curve fits must survive no-arbitrage drift: only the Segal–Bargmann kernel family keeps kernel ridge regression states drift-invariant

The kernel ridge regression estimate of the discount curve lies in a finite-dimensional affine space determined only by the kernel and the cash flow dates, and the kernels for which this space stays invariant under the risk-neutral drift are now fully classified. Kernel choices for curve fitting had been judged mainly by smoothness rewards for static estimation, while a continuous-time arbitrage-free model requires the curve drift to equal the short rate times the curve; the two had not been examined under one condition. The classification covers every grid of cash flow dates: apart from kernels whose Hilbert space has dimension at most two, the drift-invariant kernels are exactly the Segal–Bargmann family, whose kernel is given by a scale parameter and an exponential term.

For a qualifying kernel, the estimated discount curve is an affine combination of flat discount curves, each corresponding to a constant continuously compounded yield. The levels of these flat curves are tied to the cash flow date grid, so the estimate carries an explicit financial interpretation rather than coefficients of basis functions with no rate meaning. On a fixed grid the basis functions are eigenfunctions of the kernel operator with distinct eigenvalues and are therefore linearly independent; drift-invariant subspaces can only be spanned by those basis functions that correspond to a sub-grid.

Inside this hypothesis space, the coefficients of a discount curve model have quadratic drift under the risk-neutral measure, while the discounted coefficients have linear drift. The linear drift places the model in the linear-rational term structure class and yields well-defined multi-factor models once the volatility of the discounted coefficients is specified. The drift condition is equivalent to discounted bond prices being local martingales; if the state price density is positive, discounted bond prices are true martingales and bond prices are strictly positive.

The discussion is limited to kernel ridge regression estimates of the discount curve and their invariance under the risk-neutral drift, with kernels continuous and continuously differentiable in each variable; the conclusions cover every grid of cash flow dates, but the authors explicitly do not study empirical performance.

The authors leave two questions open: how the kernel performs empirically in curve estimation, and how sub-grid selection compares with other dimension reductions; whether a weaker form of dynamic consistency can be combined with a smoothness reward is left for future work.