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As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is adapted to its filtration, I can write the swap rate as an ito integral w.r.t this Brownian Motion.

If risk neutral (money market numeraire) dynamics are one factor hull white, and thus swap rate is adapted to the filtration of this Brownian motion, can I not apply Girsanov's theorem to conclude that there is a brownian motion in annuity measure (with the same filtration as brownian motion in RN measure, since it only carries an extra drift term) under which swap rate is measurable?

Thus, can I write the swap rate to evolve as an ito integral in the annuity measure without loss of generality?

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