\begin{align}
\nonumber \partial_tV_i+\sup_{\pi_i}\bigg\{&\sum_{j\neq i}\left(rx_j+(\theta(p)-r)\pi^*_j\right)\partial_{x_j}V_i+\left(rx_i+(\theta(p)-r)\pi_i\right)\partial_{x_i}V_i\\
\nonumber&+\frac{\sigma^2}{2}\sum_{j\neq i}\sum_{k\neq i}\pi_j^*\pi^*_k\partial_{x_jx_k}V_i+\frac{\sigma^2}{2}\pi_i^2\partial_{x_ix_i}V_i+{\sigma^2}\pi_i\sum_{j\neq i}\pi_j^*\partial_{x_ix_j}V_i\\
\nonumber&+\eta(p)\partial_pV_i+\frac{\beta(p)^2}{2}\partial_{pp}V_i+ \sigma\beta(p)\sum_{j\neq i}\pi^*_j\partial_{x_jp}V_i+\sigma\pi_i\beta(p)\partial_{x_ip}V_i\\
\nonumber &-\frac{\gamma_i\sigma^2}{2}\sum_{j\neq i}\sum_{k\neq i}{\pi_j^*\pi_k^*}\partial_{x_j}g_i\partial_{x_k}g_i\\
\nonumber &-\frac{\gamma_i\sigma^2}{2}\pi_i^2 (\partial_{x_i}g_i)^2-\gamma_i\sigma^2\pi_i\sum_{j\neq i}\pi_j^*\partial_{x_i}g_i\partial_{x_j}g_i\\
&-\frac{\gamma_i\beta(p)^2}{2} (\partial_{p}g_i)^2-\gamma_i\sigma\pi_i\beta(p)\partial_{x_i}g_i\partial_{p}g_i-\gamma_i\sigma\beta(p)\sum_{j\neq i}\pi_j^*\partial_{p}g_i\partial_{x_j}g_i\bigg\}=0
\label{V_i}
\end{align}
