Continuity Properties of Game-theoretic Upper Expectations

TL;DR

This paper investigates the mathematical properties of game-theoretic upper expectations in discrete-time uncertain processes, establishing their fundamental and advanced continuity characteristics.

Contribution

It proves key properties and advanced continuity results of game-theoretic upper expectations, extending their theoretical understanding.

Findings

Proved basic properties like monotonicity and law of iterated expectations.

Established continuity with respect to non-decreasing sequences.

Demonstrated continuity for sequences of finitary measurable functions.

Abstract

We consider discrete-time uncertain processes with finite state space and study the properties of game-theoretic upper expectations developed by Shafer and Vovk. We start by proving some basic properties, e.g. monotonicity, law of iterated upper expectations,... Afterwards, we show that this operator satisfies several, more involved continuity properties, including continuity with respect to non-decreasing sequences and continuity with respect to specific sequences of so-called `finitary measurable' functions, which are functions that depend on a finite number of states.