Consider two zero-coupon bond prices of the form (P(t, T)=Fleft(t, r_{t} ight)) and (P(t, S)=Gleft(t, r_{t} ight)),

Consider two zero-coupon bond prices of the form \(P(t, T)=F\left(t, r_{t}\right)\) and \(P(t, S)=G\left(t, r_{t}\right)\), where \(\left(r_{t}\right)_{t \in \mathbb{R}_{+}}\)is a short-term interest rate process. Taking \(N_{t}:=P(t, T)\) as a numéraire defining the forward measure \(\widehat{\mathbb{P}}\), compute the dynamics of \((P(t, S))_{t \in[0, T]}\) under \(\widehat{\mathbb{P}}\) using a standard Brownian motion \(\left(\widehat{W}_{t}\right)_{t \in[0, T]}\) under \(\widehat{\mathbb{P}}\).