I am reading this article: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3330240 and am trying to figure out what is written in the appendix, pages 22 and 23 They wrote:
\begin{align*} R_j(t) &= \frac{1}{\tau_j}\left[\frac{P(t,T_{j-1})}{P(t,T_{j})}-1\right] \end{align*} and they said by applying Itô's lemma we get:
\begin{align*} dR_j(t) &= \left[R_j(t) + \frac{1}{\tau_j}\right] \int_{T_{j-1}}^{T_{j}} \sigma(t,u) 1_{\{t\leq u\}}du \ dW_j(t) \end{align*}
given that
\begin{align*} \frac{dP(t,T)}{P(t,T)} = r(t)dt-\int_t^T \sigma(t,u)1_{\{t\leq u\}}du\ dW(t) \end{align*}
I tried to write \begin{align*} \frac{P(t,T_{j-1})}{P(t,T_{j})} & = P(T_{j-1},T_j)\\ dP(T_{j-1},T_j) & = r(T_{j-1})dT_{j-1}-\int_{T_{j-1}}^{T_j} \sigma(T_{j-1},u)du\ dW(T_{j-1}) \end{align*} But I am not really sure about it, because it does not lead to the result in the article. Do you have any clue?
I finally found the answer (image below). Maybe it will help someone else.