The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical Levy processes. It turns out that solutions, under rather weak requirements, do not have c`adl`ag modification. Some natural open questions are also stated.
In this paper we study the asymptotic properties of the power variations of stochastic processes of the type X=Y+L, where L is an alpha-stable Levy process, and Y a perturbation which satisfies some mild Lipschitz continuity assumptions. We establish
local functional limit theorems for the power variation processes of X. In case X is a solution of a stochastic differential equation driven by L, these limit theorems provide estimators of the stability index alpha. They are applicable for instance to model fitting problems for paleo-climatic temperature time series taken from the Greenland ice core.
