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This work defines and studies convolution kernels that preservenonnegativity. When the past dynamics of a process is integrated with aconvolution kernel like in Stochastic Volterra Equations or in the jumpintensity of Hawkes processes, this property allows to get the nonnegativity ofthe integral. We give characterizations of these kernels and show in particularthat completely monotone kernels preserve nonnegativity. We then apply theseresults to analyze the stochastic invariance of a closed convex set byStochastic Volterra Equations. We also get a comparison result in dimensionone. Last, when the kernel is a positive linear combination of decayingexponential functions, we present a second order approximation scheme for theweak error that stays in the closed convex domain under suitable assumptions.We apply these results to the rough Heston model and give numericalillustrations.

Nonnegativity preserving convolution kernels. Application to Stochastic Volterra Equations in closed convex domains and their approximation