$α$-scaled strong convergence and stability of stochastic theta method for time-changed stochastic differential equations with local Lipschitz coefficients

Abstract: We propose the first $\alpha$-parameterized framework for solving time-changed stochastic differential equations (SDEs), explicitly linking convergence rates and key driving parameter of the underlying stochastic processes. Theoretically, we derive exact moment estimates and exponential moment estimate 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 critical gap in traditional methods that treat these factors independently. We also investigate the criteria of asympotical mean square stability of the ST method. Finally, some numerical simulations are presented to illustrate the theoretical results.