On non-negative solutions of stochastic Volterra equations with jumps and non-Lipschitz coefficients

TL;DR

This paper establishes conditions for the existence and uniqueness of non-negative solutions to stochastic Volterra equations with jumps and non-Lipschitz coefficients, extending applications to finance such as alpha-stable Cox--Ingersoll--Ross processes.

Contribution

It introduces new conditions ensuring strong existence and pathwise uniqueness of solutions for stochastic Volterra equations with jumps, including a Volterra extension of the alpha-stable CIR process.

Findings

Proved strong existence and pathwise uniqueness under specific conditions.

Extended the alpha-stable CIR process to Volterra type.

Applied results to Levy-driven stochastic Volterra equations.

Abstract

We consider one-dimensional stochastic Volterra equations with jumps for which we establish conditions upon the convolution kernel and coefficients for the strong existence and pathwise uniqueness of a non-negative c\`adl\`ag solution. By using the approach recently developed in arXiv:2302.07758, we show the strong existence by using a nonnegative approximation of the equation whose convergence is proved via a variant of the Yamada--Watanabe approximation technique. We apply our results to L\'evy-driven stochastic Volterra equations. In particular, we are able to define a Volterra extension of the so-called alpha-stable Cox--Ingersoll--Ross process, which is especially used for applications in Mathematical Finance.