Maximal inequalities and exponential estimates for stochastic convolutions driven by L\'{e}vy-type processes in Banach spaces with application to stochastic quasi-geostrophic equations

We present remarkably simple proofs of Burkholder-Davis-Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by L'{e}vy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of It^{o}'s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with L'{e}vy noise is established.