On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient
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| Format: | Preprint |
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2025
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| author | Müller-Gronbach, Thomas Rauhögger, Christopher Yaroslavtseva, Larisa |
| author_facet | Müller-Gronbach, Thomas Rauhögger, Christopher Yaroslavtseva, Larisa |
| contents | We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient. More precisely, we essentially assume that the drift coefficient is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$ that is an orientable $C^4$-hypersurface of positive reach, the diffusion coefficient is assumed to be Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and the diffusion coefficient has a non-degenerate portion orthogonal to $Θ$.
In recent years, a number of results have been proven in the literature for strong approximation of such SDEs and, in particular, the performance of the Euler-Maruyama scheme was studied. For $d=1$ and finite $Θ$ it was shown that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\geq 1$ as in the classical case of Lipschitz continuous coefficients. For $d>1$, it was only known so far, that the Euler-Maruyama scheme achieves an $L_2$-error rate of at least $1/4-$ if, additionally, the coefficients $μ$ and $σ$ are globally bounded.
In this article, we prove that in the above setting the Euler-Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2-$ for all $d\in\mathbb{N}$ and all $p\geq 1$. The proof of this result is based on the well-known approach of transforming such an SDE into an SDE with globally Lipschitz continuous coefficients, a new Itô formula for a class of functions which are not globally $C^2$ and a detailed analysis of the expected total time that the actual position of the time-continuous Euler-Maruyama scheme and its position at the preceding time point on the underlying grid are on 'different sides' of the hypersurface $Θ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01630 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient Müller-Gronbach, Thomas Rauhögger, Christopher Yaroslavtseva, Larisa Numerical Analysis Probability We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient. More precisely, we essentially assume that the drift coefficient is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$ that is an orientable $C^4$-hypersurface of positive reach, the diffusion coefficient is assumed to be Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and the diffusion coefficient has a non-degenerate portion orthogonal to $Θ$. In recent years, a number of results have been proven in the literature for strong approximation of such SDEs and, in particular, the performance of the Euler-Maruyama scheme was studied. For $d=1$ and finite $Θ$ it was shown that the Euler-Maruyama scheme achieves an $L_p$-error rate of at least $1/2$ for all $p\geq 1$ as in the classical case of Lipschitz continuous coefficients. For $d>1$, it was only known so far, that the Euler-Maruyama scheme achieves an $L_2$-error rate of at least $1/4-$ if, additionally, the coefficients $μ$ and $σ$ are globally bounded. In this article, we prove that in the above setting the Euler-Maruyama scheme in fact achieves an $L_{p}$-error rate of at least $1/2-$ for all $d\in\mathbb{N}$ and all $p\geq 1$. The proof of this result is based on the well-known approach of transforming such an SDE into an SDE with globally Lipschitz continuous coefficients, a new Itô formula for a class of functions which are not globally $C^2$ and a detailed analysis of the expected total time that the actual position of the time-continuous Euler-Maruyama scheme and its position at the preceding time point on the underlying grid are on 'different sides' of the hypersurface $Θ$. |
| title | On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient |
| topic | Numerical Analysis Probability |
| url | https://arxiv.org/abs/2504.01630 |