Principal Components Analysis for Semimartingales and Stochastic PDE

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.