Deep ReLU neural networks overcome the curse of dimensionality for partial integrodifferential equations

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