Classification of Stochastic Systems: Deep Learning and Hypothesis Testing

This paper is devoted to classification of stochastic systems given by stochastic differential equations in continuous time. We develop two novel approaches. The first one is based on the use of a deep neural network (NN), whereas the second one uses hypothesis test-based methods. The idea of deep learning method focuses on treating the given stochastic system models by generating Monte Carlo sample paths. These samples are used to train a deep neutral network. A least square error is used as the loss function for network training. Then the resulting weights are applied to out of sample Monte Carlo paths for testing. The underlying problem is then converted to a stochastic optimization task. Recursive stochastic algorithms are developed; convergence of the algorithm and rate of convergence are fully analyzed. Such deep NN approach compares favorably to the hypothesis test approaches. Then mean reversion models are studied to show the adaptiveness and power of our deep NN method. An advantage of the deep NN approach is real data can be used directly to train the deep neutral network. Therefore, model calibration can be by-passed all together.