Integrability and regularity of the flow of stochastic differential equations with jumps

We derive sufficient conditions for the differentiability of all orders for the flow of stochastic differential equations with jumps and prove related $L^p$-integrability results for all orders. Our results extend similar results obtained by H. Kunita [Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in Real and Stochastic Analysis, Birkhäuser Boston, 2004, pp. 305–373] for first order differentiability and rely on the Burkholder–Davis–Gundy (BDG) inequality for time inhomogeneous Poisson random measures on $\mathbf{R}_+\times \mathbf{R}$, for which we provide a new proof.