Strong and weak convergence orders of numerical methods for SDEs driven by time-changed Lévy noise

TL;DR

The work addresses numerical approximation of SDEs driven by time-changed Lévy noise under global Lipschitz assumptions. It leverages a duality principle to relate time-changed processes to original SDEs and derives a strong convergence rate of $\tfrac{1}{2}$ for the stochastic $\theta$ method with $\theta \in [0,1]$ when paired with inverse-subordinator simulation, and a weak convergence rate of 1 for Euler–Maruyama via Kolmogorov backward PIDEs. Theoretical results are complemented by numerical experiments in a 1D setting, which corroborate the predicted rates. The findings provide rigorous convergence guarantees for two common numerical schemes in the context of time-changed Lévy noise, with implications for simulations of irregular-temporal stochastic systems. Future work will extend the analysis to non-globally Lipschitz coefficients and explore large deviations principles for these time-changed models.

Abstract

This work investigates the strong and weak convergence orders of numerical methods for SDEs driven by time-changed Lévy noise under the globally Lipschitz conditions. Based on the duality theorem, we prove that the numerical approximation generated by the stochastic $θ$ method with $θ\in [0,1]$ and the simulation of inverse subordinator converges strongly with order $1/2$. Moreover, the numerical approximation combined with the Euler--Maruyama method and the estimate of inverse subordinator is shown to have the weak convergence order $1$ by means of the Kolmogorov backward partial integro differential equations. These theoretical results are finally confirmed by some numerical experiments.

Figures (5)

• Figure 1: Sample paths for $\{D(t)\}_{t \in [0,1]}$ with $\Delta = 2^{-8}$
• Figure 2: Sample paths for $\{E_ {\Delta}(t)\}_{t \in [0,1]}$ with $\Delta = 2^{-9}$
• Figure 3: Sample paths for $\{X_{\Delta}(t)\}_{t \in [0,1]}$ with $\Delta = 2^{-15}$
• Figure 4: Strong convergence order of stochastic $\theta$ method
• Figure 5: Weak convergence order of Euler--Maruyama method

Theorems & Definitions (21)

• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 3.1
• proof
• Lemma 3.3
• proof
• Theorem 3.4