Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations
Lukas Gonon, Christoph Schwab
Abstract
Deep neural networks (DNNs) with ReLU activation function are proved to be able to express viscosity solutions of linear partial integrodifferential equations (PIDEs) on state spaces of possibly high dimension d. Admissible PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection, and for pure jump Lévy processes. We prove for such PIDEs arising from a class of jump-diffusions on [Formula: see text], that for any suitable measure [Formula: see text] on [Formula: see text], there exist constants [Formula: see text] such that for every [Formula: see text] and for every [Formula: see text] the DNN [Formula: see text]-expression error of viscosity solutions of the PIDE is of size [Formula: see text] with DNN size bounded by [Formula: see text]. In particular, the constant [Formula: see text] is independent of [Formula: see text] and of [Formula: see text] and depends only on the coefficients in the PIDE and the measure used to quantify the error. This establishes that ReLU DNNs can break the curse of dimensionality (CoD for short) for viscosity solutions of linear, possibly degenerate PIDEs corresponding to suitable Markovian jump-diffusion processes. As a consequence of the employed techniques, we also obtain that expectations of a large class of path-dependent functionals of the underlying jump-diffusion processes can be expressed without the CoD.