Generative model for fBm with deep ReLU neural networks.

Summary We provide a large probability bound on the uniform approximation of fractional Brownian motion $(B^H(t) : t ∈ [0,1])$ with Hurst parameter $H$, by a deep-feedforward ReLU neural network fed with a $N$-dimensional Gaussian vector, with bounds on the network construction (number of hidden layers and total number of neurons). Essentially, up to log terms, achieving an uniform error of $\mathcal{O}(N^{-H})$ is possible with log$(N)$ hidden layers and $\mathcal{O} (N )$ parameters. 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 $(H \ne 1/2)$, on the Lemarié-Meyer wavelet representation of $B^H$. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.