Strong Convergence of Fully Discrete Tamed Euler Approximations for Time-Changed Stochastic Delay Differential Equations

We establish the strong convergence of a fully discrete equidistant tamed Euler–Maruyama (ETEM) method for time-changed stochastic delay differential equations (TCSDDEs) with state-dependent $\mathrm{d}t$ drifts. A coupled one-sided condition on the subordinator-driven drift and diffusion allows superlinear coefficients with polynomial growth in both the current and delayed states. For subordinators whose Laplace exponent satisfies $\psi (\lambda )\ge c_1{\lambda }^{\alpha }$ for all sufficiently large $\lambda$, with $\alpha \in (0,1)$, we derive strong error bounds in terms of the time step $\mathrm{\Delta }$ and the subordinator sampling step $\delta$, without imposing a relationship between them. Comparing the partitions generated by the exact and approximate inverse subordinators controls the sampling error without introducing a factor proportional to the number of time steps. For coefficients Lipschitz in both time arguments and Lipschitz history, the scheme achieves a strong convergence rate of order ${\mathrm{\Delta }}^{\alpha \mathrm{/}2}(1+\mathrm{\mid }\log \mathrm{\Delta }\mathrm{\mid }{)}^{(1-\alpha )\mathrm{/}2}+\sqrt{\delta (1+\mathrm{\mid }\log \delta \mathrm{\mid })}$. A terminal lower bound $c({\mathrm{\Delta }}^{\alpha \mathrm{/}2}+{\delta }^{1\mathrm{/}2})$ for a multiplicative-noise equation with a stable subordinator establishes the sharpness of both polynomial exponents for this Euler scheme. Numerical experiments illustrate convergence for nonautonomous and multidimensional superlinear delay equations, as well as delay equations with low-regularity time dependence.