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 L² sense. Our stochastic Milstein-scheme can achieve the desired accuracy O(ε), and the computational cost of a single sample from O(N²+ε-2-2/α) and O(N+ε-1-1/α) to O(Nlog N+ε-2/α) and O(log N+ε-1-1/α) when T1, and for T≈1, the reductions are to O(Nlog² N+ε-2/α) and O(log ²N+ε-1-1/α), 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 L²-norm framework.