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

## TL;DR

This paper establishes conditions under which the flow of stochastic differential equations with jumps is infinitely differentiable and $L^p$-integrable, extending previous first-order results using new inequalities for Poisson measures.

## Contribution

It provides new sufficient conditions for higher-order differentiability and integrability of stochastic flows with jumps, including a novel proof for inequalities related to Poisson random measures.

## Key findings

- Derived conditions for infinite differentiability of stochastic flows with jumps.
- Proved $L^p$-integrability results for all orders of derivatives.
- Introduced a new proof for inequalities involving Poisson random measures.

## Abstract

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 in [Kun04] for first order differentiability and rely on the Burkholder-Davis-Gundy inequality for time inhomogeneous Poisson random measures on ${\Bbb R}_+\times {\Bbb R}$, for which we provide a new proof.