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Euler scheme for SDEs driven by fractional Brownian motions: integrability and convergence in law
- Publication Year :
- 2023
-
Abstract
- In this note we consider stochastic differential equations driven by fractional Brownian motions (fBm) with Hurst parameter $H>1/3$. We prove that the corresponding modified Euler scheme and its Malliavin derivatives are integrable, uniformly with respect to the step size $n$. Then we use the integrability results to derive the convergence rate in law $n^{1-4H+\varepsilon} $ for the Euler scheme. The proof for integrability is based on a nontrivial generalization (to quadratic functionals of the fBm) of a now classical greedy sequence argument laid out by Cass, Litterer and Lyons. The proof of weak convergence applies Malliavin calculus and some upper-bound estimates for weighted random sums.<br />This is a companion paper to arXiv:2305.10365. We apologize for the text overlap, due to some common preliminary notions
- Subjects :
- Probability (math.PR)
FOS: Mathematics
Mathematics - Probability
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....4142be38e3dd17bae96ba3361f269bfb