A de Finetti-type theorem for random-rotation-invariant continuous semimartingales

TL;DR

This paper characterizes continuous semimartingales invariant under predictable random rotations, showing they can be represented as integrals of predictable processes with respect to independent Brownian motions.

Contribution

It provides a de Finetti-type theorem for a class of semimartingales, extending invariance principles to the setting of random rotations.

Findings

Semimartingales invariant under predictable rotations are representable via Brownian motion integration.

The result generalizes classical de Finetti theorems to continuous semimartingales.

The characterization aids in understanding symmetry properties in stochastic processes.

Abstract

We provide a characterization of continuous semimartingales whose law is invariant with respect to predictable random rotations. In particular we prove that all such semimartingales are obtained by integrating a predictable process with respect to an independent $n$ dimensional Brownian motion.