In interest modelling we ahve the the simply compounded forward rate at time t over the interval $[T,S]$, where $t<T<S$ is defined as
$$1+f(t,T,S)(S-T)=\frac{P(t,T)}{P(t,S)}.$$
$P(t,T)$ is the price of a zero coupon bond at time $t$ with maturity $T$.
If $r$ is the short-rate, then we have that
$$P(t,T)=E_Q\left[\exp\left(-\int\limits_t^T r_sds\right)\Bigg|\mathcal{F}_t\right],$$ for some suitable martingale measure $Q$, likewise for $P(s,T)$.
This means that $$1+f(t,T,S)(S-T)=\frac{E_Q\left[\exp\left(-\int\limits_t^T r_sds\right)\Bigg|\mathcal{F}_t\right]}{E_Q\left[\exp\left(-\int\limits_t^S r_sds\right)\Bigg|\mathcal{F}_t\right]}.$$
It is now natural to look at the quantity $P(T,S)$. And we have
$$P(T,S)=E_Q\left[\exp\left(-\int\limits_T^S r_sds\right)\Bigg|\mathcal{F}_T\right].$$
What I am wondering is if there is a connection between $P(S,T)$ and $f(t,T,S)$?