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

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 Added by Nicolas Privault
 Publication date 2019
  fields
and research's language is English




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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.



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Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps. A natural extension to this general setting of reduction and reconstruction theory for symmetric SDEs is provided. Our theorems imply as special cases non trivial invariance results concerning a class of affine iterated random maps as well as symmetries for numerical schemes (of Euler and Milstein type) for Brownian motion driven SDEs.
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