Can the $L^1$-$L^\infty$ duality be restored for non-dominated families of probability measures?
非受控概率测度族下能否恢复 L¹–L∞ 对偶性？AI 解读 ↓
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非受控概率测度族下能否恢复 L¹–L∞ 对偶性？
- 研究问题
- 当模型不确定性由非受控概率测度族表达、经典 L¹–L∞ 对偶关系失效时，能否通过扩展概率空间恢复适当的对偶结构？
- 方法与思路
- 作者构造原 σ 代数的规范扩展，使拟处处有界函数空间与相应有限符号测度空间的对偶等距同构，并联系容量方法。
- 材料陈述的贡献
- 摘要称，构造具有最小完备扩展性质，覆盖多类非受控模型、统一已有框架，并将严格无偏假设检验的刻画推广到该环境。
- 阅读关注点
- 应核对测度空间与范数的精确定义、最小扩展的适用条件、容量框架等价所需假设，以及严格无偏检验推广的范围。
- 阅读边界
- 这是 arXiv 预印本，不能据此认定已同行评审或期刊接收；摘要未列完整技术假设，金融模型例子不能替代具体应用有效性证明。
原文摘要
The duality $L^{\infty}\simeq (L^{1})'$ frequently breaks down in the presence of model uncertainty, where a single reference measure $P$ is replaced by a non-dominated family of probability measures $\mathcal{P}$. The unavailability of classical measure-theoretic and functional-analytic tools in this regime poses a significant obstacle to developing robust probabilistic frameworks. We show that this duality can be restored for a broad class of robust statistical models by extending the underlying probability space. Specifically, on the extended model, the space $\mathbb{L}^{\infty}(\mathcal{P})$ of $\mathcal{P}$-quasi-surely bounded functions is isometrically isomorphic to the dual of the space of finite signed measures absolutely continuous with respect to at least one element of $\mathcal{P}$. The proposed extension is canonical: it is the smallest $\mathcal{P}$-complete extension of the original $σ$-algebra for which $\mathbb{L}^{\infty}(\mathcal{P})$ is the dual of any normed space. Our assumptions encompass several prominent non-dominated settings, including infinite product measures, Gaussian processes, the Black-Scholes model with uncertain constant volatility and drift, robust binomial models, and, more generally, infinite sequences from any parametric model with almost surely estimable parameters. Furthermore, we unify the existing frameworks of Cohen (2012) and Liebrich et al. (2022), demonstrating that our construction is equivalent to the capacity-based approach under mild assumptions satisfied by the aforementioned examples. Finally, we apply our theory to extend Kraft's (1955) characterization of strictly unbiased hypothesis tests to non-dominated cases.