Permutation Invariant Functionals of L\'evy Processes

Abstract

We study natural invariance properties of functionals defined on L\'evy processes and show that they can be described by a simplified structure of the deterministic chaos kernels in It\^o's chaos expansion. These structural properties of the kernels relate intrinsically to a measurability with respect to invariant $\sigma$-algebras. This makes it possible to apply deterministic functions to invariant functionals on L\'evy processes while keeping the simplified structure of the kernels. This stability is crucial for applications. Examples are given as well.