Symmetries and invariance properties of stochastic differential equations driven by semimartingales with jumps

TL;DR

This paper explores the symmetry and invariance properties of stochastic differential equations driven by semimartingales with jumps, introducing new concepts of gauge and time symmetries to broaden the understanding of these systems.

Contribution

It introduces the concepts of gauge and time symmetries for semimartingales on Lie groups, expanding the class of symmetries applicable to SDEs driven by jumps.

Findings

Defined stochastic symmetries for SDEs with jumps

Provided examples of gauge and time symmetric processes

Established invariance results for iterated random maps

Abstract

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general c\`adl\`ag semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markovian and non-Markovian examples of gauge and time symmetric processes are provided. The considered set of SDEs includes affine and Marcus type SDEs as well as smooth SDEs driven by L\'evy processes. Non trivial invariance results concerning a class of iterated random maps are obtained as special cases.