Generative model for fbm with deep ReLU neural networks
Résumé
Over the last few years, a new paradigm of generative models based on neural networks have shown impressive results to simulate – with high fidelity – objects in high-dimension, while being fast in the simulation phase. In this work, we focus on the simulation of continuous-time processes (infinite dimensional objects) based on Generative Adversarial Networks (GANs) setting. More precisely, we focus on fractional Brownian motion, which is a centered Gaussian process with specific covariance function. Since its stochastic simulation is known to be quite delicate, having at hand a generative model for full path is really appealing for practical use. However, designing the architecture of such neural networks models is a very difficult question and therefore often left to empirical search. We provide a high confidence bound on the uniform approximation of fractional Brownian motion B^H(t), with Hurst parameter H, by a deep-feedforward ReLU neural network fed with a Z-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction of B^H and in the general fractional Brownian motion case ( H ≠ 1/2 ), on the Lemarié-Meyer wavelet representation of B^H. This work gives theoretical support to use, and guidelines to construct, new generative models based on neural networks for simulating stochastic processes. It may well open the way to handle more complicated stochastic models written as a Stochastic Differential Equation driven by fractional Brownian motion.