The martingale part in the semimartingale decomposition of a Brownian motion with respect to an enlargement of its filtration, is an anticipative mapping of the given Brownian motion. In analogy to optimal transport theory, we define causal transport plans in the context of enlargement of filtrations, as the Kantorovich counterparts of the aforementioned non-adapted mappings. We provide a necessary and sufficient condition for a Brownian motion to remain a semimartingale in an enlarged filtration, in terms of certain minimization problems over sets of causal transport plans. The latter are also used in order to give robust transport-based estimates for the value of having additional information, as well as model sensitivity with respect to the reference measure, for the classical stochastic optimization problems of utility maximization and optimal stopping. Our results have natural extensions to the case of general multidimensional continuous semimartingales.

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因果最优传输及其与过滤扩大和连续时间随机优化的联系

布朗运动的半鞅分解中的鞅部分相对于其过滤的扩大，是给定布朗运动的预期映射。类似于最优传输理论，我们在过滤扩大的背景下定义因果传输计划，作为上述非适应映射的 Kantorovich 对应物。我们为布朗运动在扩大过滤中保持半鞅的必要和充分条件提供了在因果传输计划集上的某些最小化问题。后者还用于为具有附加信息的价值提供基于运输的稳健估计，以及相对于参考测量的模型敏感性，用于效用最大化和最优停止的经典随机优化问题。我们的结果自然扩展到一般多维连续半鞅的情况。