# Analogue of Leibniz Rule for Stochastic Integrals

Suppose $$f(t,u)=f(0,u)+\int_0^t{\mu (w,u)dw}+\int_0^t{\sigma(w,u)dB_w}$$, where $B_w$ is a standard Brownian motion. I would like to calculus the drift and diffusion of $Y_t=-\int_t^s{f(t,u)du}$ (under sufficient conditions that guarantee all the regularities).

The problem comes from the HJM model in finance, and the answer is $$dY_t=\left(f(t,t)-\int_t^s{\mu(t,u)du}\right)dt+\left(-\int^s_t{\sigma(t,u)du}\right)dB_t$$. I am really confused about where the term $f(t,t)$ in the drift comes from. Formally, I can calculate this using Leibniz rule and get $$dY_t=-\int_t^s{d_t f(t,u)du}+f(t,t)dt$$ which is equivalent to the answer, but this calculation is not justified. And the notes I have says the answer is due to Fubini's theorem for stochastic integrals. I understand that I can use Fubini's theorem to get $$Y_t=-\int_t^sf(0,u)du-\int^t_0\int^s_t\mu(w,u)dudw-\int_0^t\int_t^s\sigma(w,u)dudB_w$$. But I don't know how this could lead to the answer, since the "drift" $\int^s_t\mu(w,u)du$ also depends on $t$, and how I could get the term $f(t,t)$.

Thank you!

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I don't know if this is still relevant, but: In fixed income modelling by Claus Munk page 57-58 he proves Leibniz rule for stochastic integrals and on page 282 he does the step you dislike (in the HJM) model. The proof of Leibniz rule is really ugly and only uses stochastic fubini, but when you got the result pretty much follows. I think it will be clear to you if you find the book. –  Henrik Dec 21 '12 at 16:28