which indicates that $\mathbb{Q}^{\phi^{\ast}}$ is actually an absolutely continuous martingale measure. Besides, we have \begin{equation}
    \mathrm{d}M^{mmv,\ast}(s)S_1(s) = M^{mmv,\ast}(s^-)S_1(s^-) \Bigg[ \left(\sigma-\phi^*\right)\mathrm{d}B(s) + \int_{-1}^{\infty} \left(q-\phi_2^{\ast}(q)-q\phi_2^{\ast}(q)\right)\widetilde{N}(\mathrm{d}s,\mathrm{d}q) \Bigg],  \notag
\end{equation} which means that  $M^{mmv,\ast}$ is a pricing   operator.  $\phi_2^{\ast}(q) \leq 1$ ensures the non-negativity of $M^{mmv,\ast}$. 
