# 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 $\mathcal{O} (N )$ 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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https://hal.archives-ouvertes.fr/hal-03237854
Contributor : Stephane Girard <>
Submitted on : Wednesday, June 16, 2021 - 9:44:32 AM
Last modification on : Friday, June 18, 2021 - 4:42:20 PM

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fBm-HAL-v2.pdf
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• HAL Id : hal-03237854, version 2

### Citation

Michaël Allouche, Stéphane Girard, Emmanuel Gobet. A generative model for fBm with deep ReLU neural networks. 2021. ⟨hal-03237854v2⟩

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