# Instantaneous forward rate within the HJM framework

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.

This is known as the classical Leibniz rule. The link sends to Wikipedia, where you can find a proof. It allows to differentiate under the integral sign. A general statement of the formula is: $$\text{d}\left(\int_{g(x)}^{h(x)}f(x,s)\text{d}s\right)=h'(x)f(x,h(x))\text{d}x-g'(x)f(x,g(x))\text{d}x+\int_{g(x)}^{h(x)}\text{d}f(x,s)\text{d}s$$

• Hello thank you for your answer, but how can you apply Leibniz in this case ? The differential of $f_t(T)$ with respect to $t$ has absolutely no meaning. – Younes S Oct 11 '19 at 9:48
• Why do you say it does not have meaning? $f(t,T)$ depends upon the value of $t$ because it delimits the domain of integration for the two integrals making up $f(t,T)$. – Daneel Olivaw Oct 11 '19 at 10:59
• Take a simple example, $\tilde{f}(t)=\int_0^tx\text{d}x$, you clearly see that: $\tilde{f}(t)=t^2/2$, for which you can take the differential, thus it does make sense. – Daneel Olivaw Oct 11 '19 at 11:01
• For $T$ fixed, $f_t(T)$ is a stochastic process, that is why it has no meaning to differentiate with respect to $t$, when we are writing $df$ it is just a notation abuse, it has no meaning itself. You can refer to this topic : math.stackexchange.com/questions/99184/… – Younes S Oct 11 '19 at 12:44
• You might interpret the notation $\text{d}(\int f)$ as an abuse of notation too, just as $\text{d}f$ is an abuse of notation, where $f$ is a stochastic process. What we are representing with this notation are integrals, which are well-defined. Check out this post on Stochastic Leibniz rule. – Daneel Olivaw Oct 11 '19 at 13:15

It is just an application of the Leibniz integral rule, written in differential form. Please see here: https://en.m.wikipedia.org/wiki/Leibniz_integral_rule

Capital T is constant, t is changing, so the second term on the right hand side is the exchange of integral and differential, the first term on the right hand side is the function value at the lower integration limit times derivative of t wrt t (which is 1), the function value at upper integration limit term that you see in the Leibniz rule is zero here because T is considered constant.

• Hi, thank you for your answer, please refer to my comment above. – Younes S Oct 11 '19 at 9:50
• Thanks, I added reply there, hope that helps! – Magic is in the chain Oct 11 '19 at 21:27