Abstract

We propose a numerical scheme for hypoelliptic diffusions as an extension of Milstein scheme that outperforms Euler-Maruyama scheme and standard Milstein scheme, though they share the same convergence rate in a weak sense. In other words, an efficient simulation method is provided for a wide class of Wiener functionals. Analytic error term for the new scheme is derived and compared with that for Euler-Maruyama scheme under non-smooth test functions. We also show a deep learning-based sampling method for more general class of diffusions. Thus, an alternative to cubature method or a new random number generator on path space is provided. Sampling points of iterated integrals on path space are automatically obtained by a deep neural network. Then, higher order schemes are implemented by means of deep learning. The effectiveness of the proposed schemes is shown through numerical experiments for diffusion models appearing in financial mathematics. The talk is based on joint works with Yuga Iguchi (University College London) and Riu Naito (Hitotsubashi University).