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We have $Y_t=e^{\int_0^t W_sds}$.

How do I obtain the dynamics of $Y_t$ (i.e. $dY_t$)? It seems that we can't use Ito Lemma because $\int_0^t W_sds$ is not in the form $X_t = \int_0 ^t \sigma_s dW_s + \int_0 ^t \mu_s ds$.

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...we can't use Ito [l]emma because $\displaystyle\int_0^t W_sds$ is not in the form $X_t = \displaystyle\int_0 ^t \sigma_s dW_s + \int_0 ^t \mu_s ds$.

Actually $X_t=\displaystyle\int_0^t W_sds$ is in the form $X_t = \displaystyle\int_0 ^t \sigma_s dW_s + \int_0 ^t \mu_s ds$ with, for every $t\geqslant0$, $$\sigma_t=0,\qquad \mu_t=W_t.$$

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