A generative model for fBm with deep ReLU neural networks
Résumé
We provide a large probability bound on the uniform approximation of fractional Brownian motion (BH(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 O(N−H) is possible with log(N) hidden layers and O(NlogN) parameters. Our analysis relies, in the standard Brownian motion case (H=1/2), on the Levy construction of BH and in the general fractional Brownian motion case (H≠1/2), on the Lemarié-Meyer wavelet representation of BH. This work gives theoretical support on new generative models based on neural networks for simulating continuous-time processes.
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