We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(t,T)&=\exp-\bigg(\int_0^Tf(0,u)du+\int_0^t\int_s^T\sigma(s,u)dudW_s+\int_0^t\int_s^T\alpha(s,u)duds\bigg)\\ &=\exp\bigg(-\int_0^Tf(0,u)du+\int_0^t\underbrace{-\int_s^T\sigma(s,u)du}_{\Sigma(s,T)}dW_s-\int_0^t\int_s^T\alpha(s,u)duds\bigg) \end{align} By Ito's Lemma $$d_tZ(t,T)=Z(t,T)\Bigg(\bigg(\frac{1}{2}\Sigma^2(t,T)-\int_t^T\alpha(t,u)du\bigg)dt+\Sigma(t,T)dW_t\Bigg)$$ My question is, how was Ito's Lemma applied to the term $\int_0^t\int_s^T\sigma(s,u)dudW_s$, which contains an integral of Brownian motion?
Claus Munk's Fixed Income Modelling proves the following stochastic Leibniz rule on page 57-58 $$Y_t=\int_t^Tf(0,u)du+\int_t^T\int_0^t\alpha(s,u)dsdu+\int_t^T\int_0^t\sigma(s,u)dW_sdu$$ $$dY_t=\bigg(\int_t^T\alpha(t,u)du-f(t,t)\bigg)dt+\bigg(\int_t^T\sigma(t,u)du\bigg)dW_t$$ however due to the different limits in the integrals, I am unable to extrapolate from Munk's example to solve my question.
Any help is appreciated.
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