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.
翻译:触摸过程是随机的变量,在某些路径空间中具有价值。 然而, 将随机随机变量降低为路径估价的随机变量, 忽略了过滤过程, 即由过程随时间传播的信息流动。 通过将过程设置在过滤上, 我们引入了一个由高排序内核嵌入( KMEs) 组成的大家庭, 将 KME 的概念概括化, 并捕捉与过滤相关的额外信息。 我们通过基于古典内核的回归方法, 得出相关最高顺序差异( MMDs) 的经验性估测器, 并证明一致性。 然后, 我们构建一个能接收标准 MMMD 测试遗漏的信息的过滤敏感多层内核双模测试。 此外, 我们利用我们更高的程序, 利用我们更高的程序, 将通用内核内核嵌嵌嵌嵌组成一个组合, 能够解决真实世界校准和最佳遏制量化融资问题( 如美国选项的定价 ) 。 最后, 我们调整现有的测试, 以有条件的独立性测试, 以测试为标准性、 结构性的、 结构性、 和性分析性分析性分析性分析性系统, 我们设计一个统一的机算。