Martingale Functional Control variates via Deep Learning

We propose black-box-type control variate for Monte Carlo simulations by leveraging the Martingale Representation Theorem and artificial neural networks. We developed several learning algorithms for finding martingale control variate functionals both for the Markovian and non-Markovian setting. The proposed algorithms guarantee convergence to the true solution independently of the quality of the deep learning approximation of the control variate functional. We believe that this is important as the current theory of deep learning functions approximations lacks theoretical foundation. However the quality of the deep learning functional approximation determines the level of benefit of the control variate. The methods are empirically shown to work for high-dimensional problems. We provide diagnostics that shed light on appropriate network architectures.