    \begin{equation}
        \mathrm{d}M^{mmv,\ast}(s) = M^{mmv,\ast}(s^-) \Bigg( -r \mathrm{d}s - \phi_1^{\ast}\mathrm{d}B(s) - \int_{-1}^{\infty} \phi_2^{\ast}(q)\widetilde{N}(\mathrm{d}s,\mathrm{d}q) \Bigg), \quad M^{mmv,\ast}(t) = 1. \label{MMV pricing operator} \notag
    \end{equation}
