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

TL;DR

This paper develops an α-aware numerical framework for time-changed SDEs driven by an inverse α-stable subordinator, linking convergence rates of the stochastic theta method to the clock index $α ∈ (0,1)$. It derives exact and exponential moment formulas for the inverse subordinator and establishes α-sensitive moment bounds for the exact solution, enabling α-dependent convergence analysis. The authors introduce Mittag-Leffler stability for both exact and numerical solutions, providing Lyapunov criteria that generalize exponential stability to subdiffusive dynamics, and they prove that the stochastic theta method preserves mean-square Mittag-Leffler stability under appropriate step-size and implicitness conditions. They also compare with a FBEM reference scheme to obtain sharp α-dependent convergence results and present numerical simulations that confirm the theoretical rates and stability behavior. Overall, the work offers a unified framework connecting subdiffusive clock dynamics, α-dependent convergence, and Mittag-Leffler stability for TCSDEs with time-space-dependent coefficients, with practical implications for accurate and stable numerical simulations.

Abstract

We propose the first $α$-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 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 $α$ 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.

Figures (8)

• Figure 1: Sample paths of a 0.9-stable subordinator $D$ and the corresponding inverse $E$.
• Figure 2: Sample paths of time-changed Black-Scholes SDE with different coefficients.
• Figure 3: Five sample paths of \ref{['Example 5.2']}.
• Figure 4: Distribution of sample mean square errors for \ref{['Example 5.1']}.
• Figure 5: Convergence order simulation for \ref{['Example 7.3']} with $\theta_{t}=0.05+0.03sin(2\pi t)$, $\sigma_{t}=0.4\times(1+0.05t)$, $\kappa=0.65$

Theorems & Definitions (37)

• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof
• Lemma 3.1
• proof
• Lemma 4.1
• proof