  Deep neural networks (DNNs) with ReLU activation function are proved to be
able to express viscosity solutions of linear partial integrodifferental
equations (PIDEs) on state spaces of possibly high dimension $d$. Admissible
PIDEs comprise Kolmogorov equations for high-dimensional diffusion, advection,
and for pure jump L\'{e}vy processes. We prove for such PIDEs arising from a
class of jump-diffusions on $\mathbb{R}^d$, that for any compact $K\subset
\mathbb{R}^d$, there exist constants $C,{\mathfrak{p}},{\mathfrak{q}}>0$ such
that for every $\varepsilon \in (0,1]$ and for every $d\in \mathbb{N}$ the
nomalized (over $K$) DNN $L^2$-expression error of viscosity solutions of the
PIDE is of size $\varepsilon$ with DNN size bounded by
$Cd^{\mathfrak{p}}\varepsilon^{-\mathfrak{q}}$.
  In particular, the constant $C>0$ is independent of $d\in \mathbb{N}$ and of
$\varepsilon \in (0,1]$ 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 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.
