within the HJM framework, the dynamics of the instantaneous forward rate are defined by:
$$f_t(T)=f_0(T) + \int_0^t\alpha_s(T)ds+\int_0^t\sigma_s(T)dW_s$$
or in differential form: $$df_t(T)=\alpha_t(T)dt+\sigma_t(T)dW_t$$
In the litterature (like Tankov, you can find the url below), it is written that: $$d\left(\int_t^Tf_t(u)du\right)= -f_t(t)dt+\int_t^Tdf_t(u)du $$ I could not find a proof and Tankov mentions it like it is trivial.
page 96 in :https://masterfinance.math.univ-paris-diderot.fr/attachments/article/47/processus_en_finance_6_7.pdf
Thank you for your help.
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