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

Aur\'elien Alfonsi and Guillaume Szulda

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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.

Date: 2024-02, Revised 2024-07
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