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

This paper introduces a novel α-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 α-stable subordinator E using Mittag-Leffler functions. The stochastic theta (ST) method is investigated for a general class of SDEs driven by 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 α of stable subordinator D, filling a gap in traditional methods that treat these factors independently. Furthermore, we propose the notion of Mittag-Leffler stability for both exact and numerical solutions of TCSDEs, and establish rigorous stability criteria under this framework. Comprehensive numerical simulations are provided to validate all theoretical findings and demonstrate the efficacy of the proposed method.