Strong convergence of Lévy-driven mixed stochastic integro-differential equations with application to the rough mixed volatility models

In this paper, we focus on investigating the strong convergence of Lévy-driven mixed stochastic integro-differential equations (L-mSIDEs) with singular kernels under the local Lipschitz and linear growth conditions. First, we transform the L-mSIDEs into an equivalent Lévy-driven mixed stochastic Volterra integral equations (L-mSVIEs) by a fractional calculus technique. Then, we rigorously analyze the existence, uniqueness, boundedness, and the continuous dependence of the analytical solutions to the L-mSVIEs. After that, we propose a modified stochastic Milstein method as a numerical solution for the L-mSVIEs by the local truncation technique and sum-of-exponentials (SOE) approximation scheme to improve the calculations effectively. Specifically, we derive the precise convergence order of this method under the same condition in the $ L^2 $ sense. Our stochastic Milstein-scheme can achieve the desired accuracy $ \mathcal{O}(\varepsilon) $, and the computational cost of a single sample from $ \mathcal{O}(N^2+\varepsilon^{-2-\frac{2}{\alpha}}) $ and $ \mathcal{O}(N+\varepsilon^{-1-\frac{1}{\alpha}}) $ to $ \mathcal{O}(N\log N+\varepsilon^{-\frac{2}{\alpha}}) $ and $ \mathcal {O}(\log N+\varepsilon^{-1-\frac{1}{\alpha}}) $ when $ T\gg1 $, and for $ T\approx1 $, the reductions are to $ \mathcal{O}(N\log^2 N+\varepsilon^{-\frac{2}{\alpha}}) $ and $ \mathcal{O}(\log ^2N+\varepsilon^{-1-\frac{1}{\alpha}}) $, respectively. To verify the accuracy and robustness of our theoretical framework, we conduct numerical experiments using rough mixed volatility models. By comparing our results with [18], we demonstrate that the current approach not only circumvents the restrictive integrability conditions imposed by singular kernels, but also achieves a rigorous convergence order in the $ L^2 $-norm framework.