    \begin{align*}
         \frac{H^{B,\mu} \!(\nu^B+\varepsilon w^B)\! -\! H^{B,\mu}\! (\nu^B)}{\varepsilon} &=\mathbb E \Bigg[ \int_0^T w^B_t \bigg\{ \Big(b-2\,a^B \Big)\bar Q^B_T - 2\,\eta^B \nu^B_t\\
         &\quad + \int_t^T \Big(b\int_\mathbb R x\, \mu_u(\d x) + \alpha_u -2\,\phi^B \bar Q^B_u  \Big) \d u\bigg\} \d t  \Bigg]   +\! \varepsilon A.
    \end{align*}
