# Weak symmetries of stochastic differential equations driven by semimartingales with jumps

## TL;DR

This paper investigates weak symmetries of stochastic differential equations driven by semimartingales with jumps, extending reduction theory and providing invariance results for various classes of SDEs and numerical schemes.

## Contribution

It introduces a general framework for stochastic symmetries of SDEs driven by cadlag semimartingales in Lie groups, extending existing theories and applications.

## Key findings

- Invariance results for affine iterated random maps.
- Symmetries for Euler and Milstein schemes for Brownian-driven SDEs.
- Extension of reduction and reconstruction theory for symmetric SDEs.

## Abstract

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.