# Random transformations and invariance of semimartingales on Lie groups

## TL;DR

This paper investigates the invariance properties of semimartingales on Lie groups under random transformations, providing explicit conditions and examples, with applications to stochastic analysis.

## Contribution

It generalizes the concept of invariance under random transformations for semimartingales on Lie groups and offers explicit criteria based on stochastic characteristics.

## Key findings

- Characterization of semimartingales invariant under random transformations
- Explicit necessary and sufficient conditions in terms of stochastic characteristics
- Examples of symmetric semimartingales and their applications

## Abstract

Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed.