Strong convergence and Mittag-Leffler stability of stochastic theta method for time-changed stochastic differential equations

We propose the first $\alpha$-parameterized framework for solving time-changed stochastic differential equations (TCSDEs), explicitly linking convergence rates to the driving parameter of the underlying stochastic processes. Theoretically, we derive exact moment estimates and exponential moment estimates of inverse $\alpha$-stable subordinator $E$ using Mittag-Leffler functions. The stochastic theta (ST) method is investigated for a class of SDEs driven by a time-changed Brownian motion, whose coefficients are time-space-dependent and satisfy the local Lipschitz condition. We prove that the convergence order dynamically responds to the stability index $\alpha$ of stable subordinator $D$, filling a gap in traditional methods that treat these factors independently. We also introduce the notion of Mittag-Leffler stability for TCSDEs, and investigate the criterion of Mittag-Leffler stability for both the exact and numerical solutions. Finally, some numerical simulations are presented to illustrate the theoretical results.