 Solution

Assume the displayed Gaussian forward-rate covariance drift restriction. Its diagonal instance givesUsing it on the two parts of the deterministic discounted exponent yieldsThe last equality uses covariance symmetry to reflect the triangular integration region across its diagonal. The independent increments of the integrated Gaussian forward-rate process and its conditional Gaussian moment-generating function then give, for $r\le s\le T$,All these prices are integrable because their exponents are Gaussian random variables with finite variance. Thus the mean restriction implies the discounted bond martingale condition, completing the equivalence of the first two descriptions.