  This work defines and studies convolution kernels that preserve
nonnegativity. When the past dynamics of a process is integrated with a
convolution kernel like in Stochastic Volterra Equations or in the jump
intensity of Hawkes processes, this property allows to get the nonnegativity of
the integral. We give characterizations of these kernels and show in particular
that completely monotone kernels preserve nonnegativity. We then apply these
results to analyze the stochastic invariance of a closed convex set by
Stochastic Volterra Equations. We also get a comparison result in dimension
one. Last, when the kernel is a positive linear combination of decaying
exponential functions, we present a second order approximation scheme for the
weak error that stays in the closed convex domain under suitable assumptions.
We apply these results to the rough Heston model and give numerical
illustrations.
