[Paper Review] Continuity of the Shafer-Vovk-Ville Operator

Natan T’Joens, Gert de Cooman+1 more|arXiv (Cornell University)|Apr 5, 2018

Risk and Portfolio Optimization9 references3 citations

TL;DR

This paper investigates the continuity properties of the game-theoretic upper expectation operator introduced by Shafer and Vovk, which generalizes Kolmogorov's measure-theoretic probability by avoiding the need to specify a probability measure on all measurable events. The key contribution is proving that this operator is continuous with respect to upward convergence of uniformly bounded below sequences and with respect to point-wise limits of two-sided cuts, while also establishing a generalized Fatou's Lemma in this framework.

ABSTRACT

Kolmogorovs axiomatic framework is the best-known approach to describing probabilities and, due to its use of the Lebesgue integral, leads to remarkably strong continuity properties. However, it relies on the specification of a probability measure on all measurable events. The game-theoretic framework proposed by Shafer and Vovk does without this restriction. They define global upper expectation operators using local betting options. We study the continuity properties of these more general operators. We prove that they are continuous with respect to upward convergence and show that this is not the case for downward convergence. We also prove a version of Fatous Lemma in this more general context. Finally, we prove their continuity with respect to point-wise limits of two-sided cuts.

Motivation & Objective

• To analyze the continuity properties of the Shafer-Vovk-Ville upper expectation operator in the absence of a full probability measure.
• To determine whether this operator preserves limits under upward and downward convergence of sequences of gambles.
• To extend classical results like Fatou’s Lemma to the game-theoretic framework.
• To investigate continuity with respect to point-wise limits of two-sided cuts of gambles.

Proposed method

• The paper defines the upper expectation operator via bounded below supermartingales, which represent global betting strategies in the game-theoretic framework.
• It uses the representation of upper expectation as the infimum of initial capital required to ensure a non-negative wealth process, over all such supermartingales.
• The analysis relies on constructing auxiliary gambles $ f_A $ and $ f_{A,B} $ that truncate the original gamble from below and above, respectively.
• It applies monotonicity and convergence properties of supermartingales to derive limit behavior of the upper expectation functional.
• The proofs leverage the structure of cylinder events and pathwise convergence in the sample space $ \Omega = \mathscr{X}^{\mathbb{N}} $.
• Key results are derived using the duality between upper and lower expectations and the conjugate representation $ \underline{\mathrm{E}}(f) = -\overline{\mathrm{E}}(-f) $.

Experimental results

Research questions

• RQ1 Is the Shafer-Vovk-Ville upper expectation operator continuous with respect to upward convergence of uniformly bounded below sequences of gambles?
• RQ2 Does the operator preserve limits under downward convergence, and if not, why?
• RQ3 Can a generalized version of Fatou’s Lemma be established in the game-theoretic framework?
• RQ4 Is the operator continuous with respect to point-wise limits of two-sided cuts of gambles?
• RQ5 What is the behavior of the upper expectation under truncation of gambles via $ f_A = \max\{f, A\} $?

Key findings

• The upper expectation operator $ \overline{\mathrm{E}}_{\mathrm{V}} $ is continuous with respect to upward convergence: if $ f_k \uparrow f $ and $ f_k $ are uniformly bounded below, then $ \overline{\mathrm{E}}_{\mathrm{V}}(f_k) \uparrow \overline{\mathrm{E}}_{\mathrm{V}}(f) $.
• The operator is not continuous with respect to downward convergence, as demonstrated by counterexamples in the framework.
• A generalized Fatou’s Lemma holds: if $ f_k \downarrow f $, then $ \varlimsup \overline{\mathrm{E}}_{\mathrm{V}}(f_k) \geq \overline{\mathrm{E}}_{\mathrm{V}}(f) $.
• The operator is continuous with respect to point-wise limits of two-sided cuts: $ \lim_{A \to -\infty} \lim_{B \to \infty} \overline{\mathrm{E}}_{\mathrm{V}}(f_{A,B}) = \overline{\mathrm{E}}_{\mathrm{V}}(f) $, where $ f_{A,B} = \max\{\min\{f, B\}, A\} $.
• For any gamble $ f $ with finite upper expectation, there exists a truncation $ f_A $ such that $ \overline{\mathrm{E}}_{\mathrm{V}}(f_A) < \infty $ for all $ A \leq A^* $, ensuring stability under lower truncation.
• The limit $ \lim_{A \to -\infty} \overline{\mathrm{E}}_{\mathrm{V}}(f_A) = \overline{\mathrm{E}}_{\mathrm{V}}(f) $ holds, which is crucial for establishing continuity under truncation.