Parameter-related strong convergence rate of an Euler's type method for time-changed stochastic differential equations

TL;DR

This paper addresses numerical approximation for time-changed SDEs driven by inverse subordinators with multiplicative noise. It introduces an equidistant-step Euler–Maruyama scheme that preserves the time-change structure and proves a strong convergence rate of $O(\Delta t^{(1+\alpha)/4})$ for $\alpha\in(1/2,1)$, highlighting explicit dependence on the stability index of the subordinator. The analysis relies on moment bounds for the inverse subordinator, a decomposition of the error, and time-changed BDG-type estimates, with a note that the rate can revert to the classical $1/2$ under additional drift or bounded increment conditions. Numerical simulations on a 2D time-changed SDE corroborate the theoretical rate and illustrate how the rate strengthens as $\alpha$ increases, confirming the practical impact of the time-change on convergence behavior.

Abstract

An Euler's type method with the equidistant step size is proposed for a class of time-changed stochastic differential equations driven by the multiplicative noise and the strong convergence rate that is related to the parameter of the time changing process is obtained. Such a observation of the convergence rate is significantly different from those existing results that employ methods with the random step size. Numerical simulations are provided to demonstrate the theoretical results.

Figures (3)

• Figure 1: The left figure is sample paths of an 0.8-stable subordinator $D$ (blue) and its inverse subordinator $E$ (red). The right figure is sample paths of an inverse 0.8-stable subordinator $E$ (blue) and the corresponding time-changed Brownian motion $B \circ E$ (red).
• Figure 2: The left figure shows the convergence rates for $\alpha=0.6$, and the right figure shows the convergence rates for $\alpha=0.8$. The blue line represents the theoretical convergence rates, and the red line represents the numerical convergence rates.
• Figure 3: Convergence rates with positive linear drift coefficient $1$ for $\alpha=0.6$ (left) and $\alpha=0.8$ (right).

Theorems & Definitions (8)

• Lemma 1
• Lemma 2
• Lemma 3
• Proposition 6
• Theorem 7
• proof
• Remark 1
• Example 1