Topics on Modeling High Dimensional Time Series Data

This thesis covers three topics. The first topic studies how to find compact state embeddings from high-dimensional Markov state trajectories, where the transition kernel has a small intrinsic rank. We propose an efficient method for learning a low dimensional state embedding with Reproducing Kernel Hilbert Space. State embedding can be used to cluster states into metastable sets, thereby identifying the slow dynamics. The second topic studies how to extract governing differential equation models from time-series and spatio-temporal dataset with neural network, the method also sheds light on downsizing the popular convolutional neural network structures while maintaining the same model accuracy. The third topic focuses on how to combine deep neural network and optimal control theory to design a trading strategy when asset prices are mean reverting with proportional transaction costs, the method is shown to outperform asymptotic optimal strategies derived from homogenization technique. Our approach overcomes curse of dimensionality of free boundary problem and is scalable towards higher dimensions