Weak solutions of Stochastic Volterra Equations in convex domains with general kernels

TL;DR

This paper proves new weak existence results for stochastic Volterra equations with general kernels, emphasizing solutions remaining in convex sets, and extends kernel classes including fractional kernels.

Contribution

It introduces an approximation scheme for SVEs with singular, non-convolution kernels and extends invariance properties to these kernels, ensuring solutions stay in convex domains.

Findings

Established weak existence of solutions in convex sets for SVEs with general kernels.

Extended the class of kernels preserving nonnegativity to non-convolution kernels.

Proved weak existence and uniqueness for nonnegative solutions with square-root diffusion and non-convolution kernels.

Abstract

We establish new weak existence results for $d$-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional non-convolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in \cite{Alfonsi23} to non-convolution kernels and show that, under suitable stochastic invariance property of a closed convex set by the corresponding Stochastic Differential Equation, there exists a weak solution of the SVE that stays in this convex set. We present a family of non-convolution kernels that satisfy our assumptions, including a non-convolution extension of the well-known fractional kernel. We apply our results to SVEs with square-root diffusion coefficients and non-convolution kernels, for which we prove the weak existence and uniqueness of a solution that stays within the nonnegative orthant. We derive a representation of the Laplace transform in terms of a non-convolution Riccati equation, for which we establish an existence result.