Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866911754942939136 |
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| author | Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté |
| author_facet | Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté |
| contents | We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt + dL_t, $$ with irregular $β$-Hölder drift, $β> 0$, driven by a Lévy process with exponent $α\in (0, 2]$. For $α\in [2/3, 2]$, we obtain strong $L_p$ and almost sure convergence rates in the entire range $β> 1 - α/2$, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of $α$. Notably, the obtained convergence rate does not depend on $p$, which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong $L_p$ convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_12926 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise Butkovsky, Oleg Dareiotis, Konstantinos Gerencsér, Máté Probability Numerical Analysis 60H10, 60H35, 60H50, 60J76 We study the strong rate of convergence of the Euler--Maruyama scheme for a multidimensional stochastic differential equation (SDE) $$ dX_t = b(X_t) \, dt + dL_t, $$ with irregular $β$-Hölder drift, $β> 0$, driven by a Lévy process with exponent $α\in (0, 2]$. For $α\in [2/3, 2]$, we obtain strong $L_p$ and almost sure convergence rates in the entire range $β> 1 - α/2$, where the SDE is known to be strongly well-posed. This significantly improves the current state of the art, both in terms of convergence rate and the range of $α$. Notably, the obtained convergence rate does not depend on $p$, which is a novelty even in the case of smooth drifts. As a corollary of the obtained moment-independent error rate, we show that the Euler--Maruyama scheme for such SDEs converges almost surely and obtain an explicit convergence rate. Additionally, as a byproduct of our results, we derive strong $L_p$ convergence rates for approximations of nonsmooth additive functionals of a Lévy process. Our technique is based on a new extension of stochastic sewing arguments and Lê's quantitative John-Nirenberg inequality. |
| title | Strong rate of convergence of the Euler scheme for SDEs with irregular drift driven by Levy noise |
| topic | Probability Numerical Analysis 60H10, 60H35, 60H50, 60J76 |
| url | https://arxiv.org/abs/2204.12926 |