    \begin{equation}\label{eq:pin_dis_hjb}
        \begin{split}
            \frac{V_{k,i,j} -U^{\star}_{k-1,i,j}  }{\Delta t} + \frac{1}{2\kappa}\left(\frac{V_{k,i+1,j} -V_{k,i-1,j}  }{2\,\Delta x } - s_j \right)^2 &+ \frac{p-s_j}{T-t_{k-1}}\,\frac{U^{\star}_{k-1,i,j+1} -U^{\star}_{k-1,i,j-1}  }{2\,\Delta s } \\ &+ \frac{\sigma^2}{2}\,\frac{U^{\star}_{k-1,i,j+1} - 2U^{\star}_{k-1,i,j} +U^{\star}_{k-1,i,j-1} }{(\Delta s)^2} = 0\;.
        \end{split}
    \end{equation}
