Suppose the forward rate is given by:
$$df(t,u)= \frac{\partial}{\partial u} \bigg( \frac{\sigma(t,u)^2}{2} \bigg)dt - \frac{\partial}{\partial u} \big( \sigma(t,u) \big)dW_t$$ where $W_t$ is a standard Brownian motion.
How do I find the dynamics of a zero coupon bond $ \ p(t,T)$?
I know that $p(t,T)= e^{-\int_t^Tf(t,s)ds}$
I was thinking of using the ito formula on $f(x,t)=e^{-x}$ (plugging in $\int_t^Tf(t,s)ds$ for $x$) but I don't know how to compute the differential of $\int_t^Tf(t,s)ds$. Any help is much appreciated!