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

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 càdlàg 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 Lévy-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.