estimation - Musiela parameterization

I have a question regarding the proof of the Musiela parametrization for the dynamics of the forward rate curve. If $T$ is the maturity, $\tau=T-t$ is the time to maturity, and $dF(t,T)$ defines the dynamics of the forward rate curve, then the Musiela parametrization defines the forward rate dynamics

$$d\bar{F}(t,\tau)=dF(t,t+\tau)$$ .

My question is regarding the next step in the working of the Musiela parametrization. All of the literature I've looked at explains the next line by simply stating that a "slight variation" of Ito is applied. The line reads:

$$d\bar{F}(t,\tau)=dF(t,T)+\frac{\partial F}{\partial T}dt$$

Can someone please clarify what variation of Ito is being used here? I'm not following. The parameters to $d\bar{F}$ do not include an Ito drift/diffusion process, so why is Ito being used?

Answer

$dF(t,T)$ describes the dynamics of the rate of a particular forward contract as time $t$ moves forward to a fixed expiration $T$.

$d\bar F(t,\tau)$ describes the dynamics of the rate at a particular point on the yield curve as time moves forward.

The differential $\frac{\partial F}{\partial T}dt$ is simply the difference between holding the expiration time $T$ constant in the case of $F$ and moving it ahead with time $t$ to stay at the same point $t+\tau$ on the yield curve in the case of $\bar F$.

Somewhere underlying all this is a drift-diffusion process, but it isn't stated explicitly in your equations.

$dF(t,t+\tau)$ is a "total" differential of $F$ with respect to a simultaneous change in both its arguments. This becomes the sum of a partial differential w.r.t. change in the first argument only, $dF(t,T)$, and a partial differential w.r.t. change in the second argument only, $\frac{\partial F}{\partial T}dt$, as time moves forward.