Continuity of the shafer-Vovk-Ville operator

Kolmogorov’s 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 Fatou’s Lemma in this more general context. Finally, we prove their continuity with respect to point-wise limits of two-sided cuts

A Particular Upper Expectation as Global Belief Model for Discrete-Time Finite-State Uncertain Processes

To model discrete-time finite-state uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that is the most conservative one under a set of basic axioms. Our motivation for these axioms, which describe how local and global belief models should be related, is based on two possible interpretations for an upper expectation: a behavioural one similar to Walley's, and an interpretation in terms of upper envelopes of linear expectations. We show that the most conservative upper expectation satisfying our axioms, that is, our model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications: it guarantees that there is a unique most conservative global belief model satisfying our axioms; and it shows that Shafer and Vovk's model can be given an axiomatic characterisation and thereby provides an alternative motivation for adopting this model, even outside their game-theoretic framework. Finally, we relate our model to the upper expectation resulting from a traditional measure-theoretic approach. We show that this measure-theoretic upper expectation also satisfies the proposed axioms, which implies that it is dominated by our model or, equivalently, the game-theoretic model. Moreover, if all local models are precise, all three models coincide.Comment: Extension of the conference paper `In Search of a Global Belief Model for Discrete-Time Uncertain Processes

Functional calculus for cadlag paths and applications to model-free finance

This thesis synthesise my research on analysis and control of path-dependent random systems under uncertainty. In the first chapter, we revisit Foellmer's concept of pathwise quadratic variation for a cadlag path and show that his definition can be reformulated in terms of convergence of quadratic sums in the Skorokhod topology. This new definition is simpler and amenable to define higher order variation for a cadlag path. In the second chapter, we introduced a new topology for functionals and adopted an abstract formulation of Functional calculus on generic domain based on the differentials introduced by Dupire (2009), Cont & Fournie (2010). Our aim is not to generalise an existing rich theory for irregular paths e.g. Lyons (1998), Friz & Hairer (2014) but to introduce a bespoke and yet versatile calculus for causal random system in general and mathematical finance in particular, in order to solve problems practically as well as bring in new aspects under uncertainty. In the final chapter, we apply functional calculus to study mathematical finance under uncertainty. We first show that every self-financing portfolio can be represented by a pathwise integral and that every generic market is arbitrage free, a fundamental property that is linked to the solution, which is characterised by a fully non-linear path dependent equation, to the optimal hedging problem under uncertainty. In particular, we obtain explicit solution for the Asian option.Open Acces

Continuous-time trading and the emergence of probability

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Deep anytime-valid hypothesis testing

We propose a general framework for constructing powerful, sequential hypothesis tests for a large class of nonparametric testing problems. The null hypothesis for these problems is defined in an abstract form using the action of two known operators on the data distribution. This abstraction allows for a unified treatment of several classical tasks, such as two-sample testing, independence testing, and conditional-independence testing, as well as modern problems, such as testing for adversarial robustness of machine learning (ML) models. Our proposed framework has the following advantages over classical batch tests: 1) it continuously monitors online data streams and efficiently aggregates evidence against the null, 2) it provides tight control over the type I error without the need for multiple testing correction, 3) it adapts the sample size requirement to the unknown hardness of the problem. We develop a principled approach of leveraging the representation capability of ML models within the testing-by-betting framework, a game-theoretic approach for designing sequential tests. Empirical results on synthetic and real-world datasets demonstrate that tests instantiated using our general framework are competitive against specialized baselines on several tasks

An Introduction to the Calibration of Computer Models

In the context of computer models, calibration is the process of estimating unknown simulator parameters from observational data. Calibration is variously referred to as model fitting, parameter estimation/inference, an inverse problem, and model tuning. The need for calibration occurs in most areas of science and engineering, and has been used to estimate hard to measure parameters in climate, cardiology, drug therapy response, hydrology, and many other disciplines. Although the statistical method used for calibration can vary substantially, the underlying approach is essentially the same and can be considered abstractly. In this survey, we review the decisions that need to be taken when calibrating a model, and discuss a range of computational methods that can be used to compute Bayesian posterior distributions