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

TL;DR

This paper introduces nonnegativity-preserving convolution kernels, characterizes their properties, and applies them to analyze stochastic invariance and approximation schemes in stochastic Volterra equations, with practical applications to the rough Heston model.

Contribution

It provides a characterization of nonnegativity-preserving kernels, analyzes stochastic invariance in Volterra equations, and develops a second order approximation scheme applicable to the rough Heston model.

Findings

Completely monotone kernels preserve nonnegativity.

The approximation scheme maintains the domain constraints.

Numerical illustrations demonstrate practical effectiveness.

Abstract

This work defines and studies one-dimensional 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.