Strong Convergence Rates for Euler Schemes of Levy-Driven SDE using Dynamic Cutting

TL;DR

This work addresses the challenge of simulating Lévy-driven SDEs with infinite activity by deriving strong $L^p$ convergence rates for Euler-type schemes using a dynamic cutting (DC) approach with a time-dependent jump threshold. The authors develop two DC-based schemes: Scheme 2, which Gaussian-approximates the small jumps, and Scheme 1, which omits them, and they prove explicit non-asymptotic convergence bounds that are robust to jump activity (characterized by $\alpha\in(0,2)$). They compare DC against the fixed-threshold Asmussen–Rosiński (AR) method and demonstrate, both theoretically and via numerical experiments, that DC yields lower errors, especially as jump activity increases. The results are supported by detailed simulation algorithms and numerical examples showing DC maintains accuracy with an intensity of large jumps scaling as $O(n)$, independent of the Lévy tail, which has practical implications for reliable simulation of high-activity Lévy processes in finance and physics.

Abstract

We derive strong Lp convergence rates for the Euler-Maruyama schemes of Levy-driven SDE using a new dynamic cutting (DC) method with a time-dependent jump threshold. In addition, we present results from numerical simulations comparing the DC and Asmussen-Rosinski (AR) approaches. These simulations demonstrate the superior accuracy achieved by the DC method.

Figures (5)

• Figure 1: AR vs DC
• Figure 2: Grid and Jump Times
• Figure 3: Strong errors for $\alpha = 0.5$ using AR and DC methods.
• Figure 4: Strong errors for $\alpha = 1.0$ using AR and DC methods.
• Figure 5: Strong errors for $\alpha = 1.5$ using AR and DC methods.

Theorems & Definitions (18)

• Theorem 4.1
• Remark 4.1
• Remark 5.1
• Lemma A.1
• proof
• Lemma A.2
• proof
• Lemma A.3
• proof
• Lemma A.4