Assume a HJM framework with the same Brownian motion driving the dynamics for every tenor. $$ df(t,T) = \alpha(t, T)dt + \sigma(t,T) dw_t \,, $$ with $\alpha(t, T) = \sigma(t,T)\int_t^T \sigma(t,s)ds$.
It can be proved that: $$ -\ln(P(t, T)) = \int_0^T f(0, u) du + \int_0^t \int_s^T \alpha(s, u) du ds + \int_0^t \int_s^T \sigma(s, u) du dw_s - \int_0^t r(u) du . $$ Define the yield for a fixed maturity $$ Y_\tau(t) := Y(t, t+\tau) = -\frac{\ln(P(t, t+\tau))}{\tau} . $$ I would like to write the SDE of this process.
This reduces to a problem having a stochastic process defined by: $$ X_t = \int_0^t h(s,t) dw_s $$ For $h$ a "well behaved" function.
Is there a way to apply Ito's lemma or any other similar method to get the corresponding SDE $dX_t$?
I am also happy to take any other approach to solve the main problem, i.e. find $dY_\tau(t)$.
I am aware that the solution is not simply as for a function depending only of $s$: $$ dX_t = h(t,t) dw_t. $$ Also I realize that if the function $h(s, t)$ is separable then I can simply take out the part depending on $t$ and apply Ito. For example for $X_t=tw_t$ were one can easily get $dX_t = w_tdt + tdw_t$, which can also be verified to be the correct answer by integrating.
However this is not the case and the function is far from separable.
I have seen this and this slightly related questions, and this proof but the lack of bibliography does not give me enough confidence to apply the result. Is there any reference I could use?