Strong uniform Wong--Zakai approximations of Lévy-driven Marcus SDEs

2025/01/31 by Pavlyukevich, Ilya, Thipyarat, Sooppawat · 1 citation
#60G51 #60H10 #60H35 #65C30 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2501.19175

Abstract

For a solution X of a Lévy-driven d-dimensional Marcus (canonical) stochastic differential equation, we show that the Wong--Zakai type approximation scheme X^h has a strong convergence of order \frac12: for each T∈ [0,∞) and all x∈\mathbb R^d we have \mathbf E sup_{kh≤ T}|X_kh(x)-X^h_kh(x)|≤ C h^(1)/(2)(1+|x|), h→ 0. We also determine the rate of the locally uniform strong convergence: for each N∈(0,∞) and ε∈ (0,1) we have \mathbf Esup_{|x|≤ N}sup_{kh≤ T}|X_kh(x)-X^h_kh(x)|≤ C h^(1-ε)/(4d), h→ 0.

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• Rates of convergence for multivariate SDEs driven by Lévy processes in the small-time stable domain of attraction

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