Small deviations in p-variation norm for multidimensional Levy processes

Abstract

Let Z be an Rd-valued Levy process with strong finite p-variation for some p<2. We prove that the &#39;&#39;decompensated&#39;&#39; process Y obtained from Z by annihilating its generalized drift has a small deviations property in p-variation. This property means that the null function belongs to the support of the law of Y with respect to the p-variation distance. Thanks to the continuity results of T. J. Lyons/D. R. E. Williams, this allows us to prove a support theorem with respect to the p-Skorohod distance for canonical SDE driven by Z without any assumption on Z, improving the results of H. Kunita. We also give a criterion ensuring the small deviation property for Z itself, noticing that the characterization under the uniform distance, which we had obtained in a previous paper, no more holds under the p-variation distance.