Invariant Sublinear Expectations

TL;DR

This paper studies invariant sublinear expectations, decomposes them into components with finite periods, and characterizes their ergodic properties, showing that strongly ergodic expectations have finite periodic decompositions and their components relate to ergodic probabilities.

Contribution

It provides a novel decomposition of invariant sublinear expectations into periodic components with finite periods and characterizes their ergodic properties.

Findings

Each component of the decomposition has a finite period.

Strongly ergodic invariant sublinear expectations have finite periodic decompositions.

The limit of the p-step time means achieves the upper expectation.

Abstract

We first give a decomposition for a $T$-invariant sublinear expectation $\mathbb{E}=\sup_{P\in\Theta}\mathrm{E}_P$, and show that each component $\mathbb{E}^{(d)}=\sup_{P\in\Theta^{(d)}}\mathrm{E}_P$ of the decomposition has a finite period $p_d\in\mathbb{N}$, i.e., \[\mathbb{E}^{(d)}\left[f-f\circ T^{p_d}\right]=0, \quad f\in\mathcal{H}.\] Then we prove that a continuous invariant sublinear expectation that is strongly ergodic has a finite period $p_{\mathbb{E}}$, and each component $\Theta^{(d)}$ of its periodic decomposition is the convex hull of a finite set of $T^{p_d}$-ergodic probabilities. As an application of the characterization, we prove an ergodicity result which shows that the limit of the $p_{\mathbb{E}}$-step time means achieves the upper expectation.