英文标题：
《Principal Components Analysis for Semimartingales and Stochastic PDE》
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作者：
Alberto Ohashi, Alexandre B Simas
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最新提交年份：
2016
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英文摘要：
  In this work, we develop a novel principal component analysis (PCA) for semimartingales by introducing a suitable spectral analysis for the quadratic variation operator. Motivated by high-dimensional complex systems typically found in interest rate markets, we investigate correlation in high-dimensional high-frequency data generated by continuous semimartingales. In contrast to the traditional PCA methodology, the directions of large variations are not deterministic, but rather they are bounded variation adapted processes which maximize quadratic variation almost surely. This allows us to reduce dimensionality from high-dimensional semimartingale systems in terms of quadratic covariation rather than the usual covariance concept.   The proposed methodology allows us to investigate space-time data driven by multi-dimensional latent semimartingale state processes. The theory is applied to discretely-observed stochastic PDEs which admit finite-dimensional realizations. In particular, we provide consistent estimators for finite-dimensional invariant manifolds for Heath-Jarrow-Morton models. More importantly, components of the invariant manifold associated to volatility and drift dynamics are consistently estimated and identified. The proposed methodology is illustrated with both simulated and real data sets.
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中文摘要：
在这项工作中，我们通过对二次变异算子引入合适的谱分析，发展了一种新的半鞅主成分分析（PCA）。受利率市场中常见的高维复杂系统的激励，我们研究了由连续半鞅生成的高维高频数据中的相关性。与传统的PCA方法相比，大变化的方向不是确定性的，而是有界变化适应过程，几乎可以肯定地使二次变化最大化。这使得我们可以根据二次协变量而不是通常的协方差概念来降低高维半鞅系统的维数。该方法允许我们研究由多维潜在半鞅状态过程驱动的时空数据。该理论被应用于允许有限维实现的离散观测随机偏微分方程。特别地，我们为Heath-Jarrow-Morton模型的有限维不变流形提供了一致估计。更重要的是，与波动性和漂移动力学相关的不变流形的组成部分得到了一致的估计和识别。所提出的方法用模拟和真实数据集进行了说明。
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分类信息：

一级分类：Mathematics        数学
二级分类：Statistics Theory        统计理论
分类描述：Applied, computational and theoretical statistics: e.g. statistical inference, regression, time series, multivariate analysis, data analysis, Markov chain Monte Carlo, design of experiments, case studies
应用统计、计算统计和理论统计：例如统计推断、回归、时间序列、多元分析、数据分析、马尔可夫链蒙特卡罗、实验设计、案例研究
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一级分类：Mathematics        数学
二级分类：Probability        概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance        数量金融学
二级分类：Computational Finance        计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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一级分类：Statistics        统计学
二级分类：Statistics Theory        统计理论
分类描述：stat.TH is an alias for math.ST. Asymptotics, Bayesian Inference, Decision Theory, Estimation, Foundations, Inference, Testing.
Stat.Th是Math.St的别名。渐近，贝叶斯推论，决策理论，估计，基础，推论，检验。
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