Could someone verify whether I have correctly applied Ito's formula in this case? The underlying topic of this is HJM drift conditions for Zero Coupon Bond prices, but this should be irrelevant.
Let $I(t,T) = -\int_t^T f(t,u)du$. Also let $d f(t,T) = \alpha(t,T)dt + \sigma(t,T) dW(t)$ for some deterministic function $\alpha, \sigma$ and a 1-d Brownian motion $W(t)$ on some underlying probability space which should be irrelevant. Applying Ito's formula on $ P(t,T) = \exp(I(t,T))$ I get
$$d P(t,T) = P_tdt + P_IdI(t) + \frac{1}{2}P_{II} d \langle I \rangle$$
where $\langle \cdot \rangle$ denotes the quadratic variation operator. Is my derivation below correct:
$$ d P(t,T) = P(t,T) f(t,t) dt - P(t,T)f(t,T)dt + \frac{1}{2} P(t,T) \sigma^2(t,T)dt$$
i.e. are my calculations
$$ d I(t) = -f(t,T)dt$$
$$ d \left< I(t) \right> = d \left< \int_0^t \sigma(u,T) dW(u), \int_0^t \sigma(u,T) dW(u), \right> = d \int_0^t \sigma^2(u,T) d \left< W(u), W(u) \right> = d \int_0^t \sigma^2(u,T) d u = \sigma^2(t,T) dt$$
correct ? I believe I have a mistake in the $dI(t)$ calculation, but I'm not sure what it is. Any help would be greatly appreciated.