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Can the differential operator be removed to get the mean/variance of an Ito process?
This is wrong! Notice that $dX_t=\mu(t,X_t)dt + \sigma(t,X_t)dW$ is a shorthand for
$$\int_0^tdX_s = \int_0^t \mu(s,X_s)ds + \int_0^t\sigma(s,X_s)dW_s$$
Integrating:
$$X_t-X_0 = \int_0^t \mu(s,X_s)d …