within the HJM framework, the dynamics of the instantaneous forward rate are defined by:
ft(T)=f0(T)+∫t0αs(T)ds+∫t0σs(T)dWs
or in differential form: dft(T)=αt(T)dt+σt(T)dWt
In the litterature (like Tankov, you can find the url below), it is written that: d(∫Ttft(u)du)=−ft(t)dt+∫Ttdft(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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