Superhedging supermartingales

IF 3 3区计算机科学 Q2 COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE

International Journal of Approximate Reasoning Pub Date : 2025-09-03 DOI:10.1016/j.ijar.2025.109567

C. Bender , S.E. Ferrando , K. Gajewski , A.L. González

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Abstract

Supermartingales are here defined in a non-probabilistic setting and can be interpreted solely in terms of superhedging operations. The classical expectation operator is replaced by a pair of subadditive operators: one defines a class of null sets, and the other acts as an outer integral. These operators are motivated by a financial theory of no-arbitrage pricing. Such a setting extends the classical stochastic framework by replacing the path space of the process by a trajectory set, while also providing a financial/gambling interpretation based on the notion of superhedging. The paper proves analogues of the following classical results: Doob's supermartingale decomposition and Doob's pointwise convergence theorem for non-negative supermartingales. The approach shows how linearity of the expectation operator can be circumvented and how integrability properties in the proposed setting lead to the special case of (hedging) martingales while no integrability conditions are required for the general supermartingale case.

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Superhedging上鞅

在这里，超鞅是在非概率设置中定义的，并且可以仅根据超对冲操作来解释。经典的期望运算符被一对子加性运算符所取代：一个定义了一个空集的类，另一个作为外积分。这些经营者的动机是无套利定价的金融理论。这种设置通过用轨迹集替换过程的路径空间扩展了经典的随机框架，同时也提供了基于超套期保值概念的金融/赌博解释。本文证明了以下经典结果的类似物：非负上鞅的Doob上鞅分解和Doob上鞅的点向收敛定理。该方法显示了期望算子的线性是如何被规避的，以及在所提出的设置中的可积性如何导致（套期）鞅的特殊情况，而一般上鞅情况不需要可积性条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

International Journal of Approximate Reasoning 工程技术-计算机：人工智能

CiteScore

6.90

自引率

12.80%

发文量

170

审稿时长

67 days

期刊介绍： The International Journal of Approximate Reasoning is intended to serve as a forum for the treatment of imprecision and uncertainty in Artificial and Computational Intelligence, covering both the foundations of uncertainty theories, and the design of intelligent systems for scientific and engineering applications. It publishes high-quality research papers describing theoretical developments or innovative applications, as well as review articles on topics of general interest. Relevant topics include, but are not limited to, probabilistic reasoning and Bayesian networks, imprecise probabilities, random sets, belief functions (Dempster-Shafer theory), possibility theory, fuzzy sets, rough sets, decision theory, non-additive measures and integrals, qualitative reasoning about uncertainty, comparative probability orderings, game-theoretic probability, default reasoning, nonstandard logics, argumentation systems, inconsistency tolerant reasoning, elicitation techniques, philosophical foundations and psychological models of uncertain reasoning. Domains of application for uncertain reasoning systems include risk analysis and assessment, information retrieval and database design, information fusion, machine learning, data and web mining, computer vision, image and signal processing, intelligent data analysis, statistics, multi-agent systems, etc.