I have a question related to this paper https://www.jstor.org/stable/2245302. Given a process for the short rate $r$, the authors consider the price process $Y$ for a consol bond that satisfies
\begin{equation} Y_t=E_t\bigg[\int_t^\infty \exp\bigg(-\int_t^sr_udu\bigg)ds\bigg]. \end{equation}
They then derive the following SDE
$$dY_t=(r_tY_t-1)dt+A(r_t,Y_t)dW_t,$$
where $A$ is some measurable function. They write "The drift $r_tY_t-1$ shown for $Y$ is implied directly by $(1.1)$ (i.e. the first equation that I wrote) and Ito's formula".
Maybe I am missing something obvious, but I am not sure how to apply Ito's lemma to derive the drift as the authors do in the paper. Could someone explain this to me? Any help would be greatly appreciated.
It should be stressed that the authors state that only the drift $r_tY_t-1$ is implied by the equation for $Y_t$ and not necessarily every diffusion coefficient $A(r_t,Y_t)$ is consistent with the equation for $Y_t$ and a certain SDE for the short rate $r_t\,.$
A way to see this is the following. The consol bond $Y_t$ is a perpetual annuity that pays continuously a coupon $ds=1\,ds\,.$ If this coupon is reinvested into the money market account $\exp(\int_0^tr_u\,du)$ then the value of that account plus $Y_t$ is a non-dividend paying asset which must be a martingale under the risk-neutral measure and in the numeraire $\exp(\int_0^tr_u\,du)\,.$ That is: with $$ \textstyle Y_t+C_t=Y_t+\int_0^t\exp(\int_s^tr_u\,du)\,ds $$ the process $$ M_t=\frac{Y_t+C_t}{\exp(\int_0^tr_u\,du)}=\textstyle Y_t\exp(-\int_0^tr_u\,du)+\int_0^t \exp(-\int_0^sr_u\,du)\,ds $$ must be a martingale. The integration-by-parts formula gives $$ dM_t=\textstyle\exp(-\int_0^tr_u\,du)\,dY_t-r_t\,Y_t\exp(-\int_0^tr_u\,du)\,dt+\exp(-\int_0^tr_u\,du)\,dt $$ which can be rewritten as $$ \textstyle\exp(\int_0^tr_u\,du)\,dM_t=dY_t-(r_t\,Y_t-1)\,dt $$ or as $$ dY_t=(r_t\,Y_t-1)\,dt+\textstyle\exp(\int_0^tr_u\,du)\,dM_t\,. $$ This is an SDE for $Y_t$ with the desired drift and an unspecified diffusion term.
Note that the SDE for a dividend paying stock can be written as $$ dS_t=(rS_t-\delta S_t)\,dt+\sigma\,S_t\,dW_t $$ which is fully consistent with the SDE for $Y_t\,.$ The difference is that the stock pays a dividend rate $\delta$ and not an absolute dividend $1\,dt\,.$