\begin{align}
\nonumber\partial_t A_i x_i&+rA_ix_i+\partial_t B_i \overline x_{(-i)}+rB_i\overline x_{(-i)}+\partial_t C_i+\eta(p)\partial_pC_i+\frac{\beta(p)^2}{2}\partial_{pp}C_i\\
\nonumber &+B_i(\theta(p)-r)\overline \pi^*_{(-i)}-\frac{\gamma_i\sigma^2}{2}\left(\frac{b_i}{N}\right)^2\sum_{j\neq i}\sum_{k\neq i}\pi_j^*\pi^*_k-\frac{\gamma_i\beta(p)}{2}(\partial_pc_i)^2-\gamma\sigma\beta(p)b_i\partial_pc_i\overline \pi^*_{(-i)}\\
&+\sup_{\pi_i}\bigg\{A_i(\theta(p)-r)\pi_i-\frac{\gamma_i\sigma^2}{2}\pi_i^2 a_i^2-\gamma_i\sigma^2a_ib_i\pi_i\overline\pi^*_{(-i)}-\gamma_i\sigma\pi_i\beta(p)a_i\partial_pc_i\bigg\}=0\label{V_i_2}
\end{align}
