Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes
Higher Order Kernel Mean Embeddings to Capture Filtrations of Stochastic Processes
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发表时间：
2021-09
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通讯作者：
C. Salvi;M. Lemercier;Chong Liu;Blanka Hovarth;T. Damoulas;Terry Lyons
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作者：
C. Salvi;M. Lemercier;Chong Liu;Blanka Hovarth;T. Damoulas;Terry Lyons
随机过程是随机变量，其值在某个路径空间中。然而，将随机过程简化为路径值随机变量忽略了它的过滤，即过程随时间携带的信息流。通过调节其过滤的过程中，我们引入了一个家庭的高阶内核均值嵌入（KME），概括了KME的概念，并捕获额外的信息相关的过滤。我们推导出经验估计相关联的高阶最大均值差异（MMD），并证明了一致性。然后，我们构建了一个过滤敏感的内核两个样本测试能够拿起信息，错过了标准的MMD测试。此外，利用我们的高阶MMD，我们构建了一个随机过程的通用内核家族，可以通过经典的基于内核的回归方法解决定量金融中的真实世界校准和最优停止问题（例如美式期权的定价）。最后，适应现有的测试条件独立的情况下，随机过程中，我们设计了一个因果发现算法，以恢复因果图的相互作用机构之间的结构依赖关系，完全从他们的多维轨迹的观察。
Stochastic processes are random variables with values in some space of paths. However, reducing a stochastic process to a path-valued random variable ignores its filtration, i.e. the flow of information carried by the process through time. By conditioning the process on its filtration, we introduce a family of higher order kernel mean embeddings (KMEs) that generalizes the notion of KME and captures additional information related to the filtration. We derive empirical estimators for the associated higher order maximum mean discrepancies (MMDs) and prove consistency. We then construct a filtration-sensitive kernel two-sample test able to pick up information that gets missed by the standard MMD test. In addition, leveraging our higher order MMDs we construct a family of universal kernels on stochastic processes that allows to solve real-world calibration and optimal stopping problems in quantitative finance (such as the pricing of American options) via classical kernel-based regression methods. Finally, adapting existing tests for conditional independence to the case of stochastic processes, we design a causal-discovery algorithm to recover the causal graph of structural dependencies among interacting bodies solely from observations of their multidimensional trajectories.