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Journal Articles Journal of Complexity Year : 2022

A generative model for fBm with deep ReLU neural networks

Abstract

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 $\mathcal{O} (N \log N)$ 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.
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Dates and versions

hal-03237854 , version 1 (26-05-2021)
hal-03237854 , version 2 (16-06-2021)
hal-03237854 , version 3 (26-01-2022)
hal-03237854 , version 4 (25-04-2022)

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Michaël Allouche, Stéphane Girard, Emmanuel Gobet. A generative model for fBm with deep ReLU neural networks. Journal of Complexity, 2022, 73, pp.101667. ⟨10.1016/j.jco.2022.101667⟩. ⟨hal-03237854v4⟩
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