The stochastic MHD equations driven by pure jump noise in $L^p$ spaces

We consider the stochastic incompressible magnetohydrodynamic equations driven by additive jump noises on either the whole space ${\mathbb{R}}^d$, $d=2,3$ or a smooth bounded domain $D$ in ${\mathbb{R}}^d$. We establish the local existence and uniqueness of a mild solution in the space $L^q(0,T;{\mathbb{L}}_{\sigma }^{p\otimes }(D))$ allowing for initial data with less regularity, including the marginal case $u_0\in {\mathbb{L}}_{\sigma }^{d\otimes }(D)$. In the two-dimensional case, we also prove the global existence of mild solutions.