Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909618732531712 |
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| author | Rauhögger, Christopher |
| author_facet | Rauhögger, Christopher |
| contents | We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient driven by a $d$-dimensional
Brownian motion $W$.
More precisely, we essentially assume that the drift coefficient $μ$ is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$
that is an orientable $C^5$-hypersurface of positive reach, the diffusion coefficient $σ$ is assumed to be
Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and $σ$ is non-degenerate.
Furthermore, both $μ$ and $σ$ are assumed to be $C^{1}$ with intrinsic Lipschitz continuous derivative on $\mathbb{R}^{d}\setminus Θ$.
We introduce, for the first time in literature, a Milstein-type method which can be used to approximate SDEs of this type for general $d \in \mathbb{N}$ and prove
that this Milstein-type scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in terms of the number of steps. This method depends, in addition to evaluations of $W$ on a fixed grid, also on iterated integrals w.r.t. components of $W$, which can in general not be represented as functionals of $W$ evaluated at finitely many time points. We additionally prove that our suggested Milstein-type method is only dependent on evaluations of $W$ on a finite, fixed grid if $σ$ is additionally commutative.
To obtain our main result we prove that a quasi-Milstein scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in our setting if $μ$ is additionally continuous, which is of interest in itself. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15509 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient Rauhögger, Christopher Probability We study strong approximation of $d$-dimensional stochastic differential equations (SDEs) with a discontinuous drift coefficient driven by a $d$-dimensional Brownian motion $W$. More precisely, we essentially assume that the drift coefficient $μ$ is piecewise Lipschitz continuous with an exceptional set $Θ\subset \mathbb{R}^d$ that is an orientable $C^5$-hypersurface of positive reach, the diffusion coefficient $σ$ is assumed to be Lipschitz continuous and, in a neighborhood of $Θ$, both coefficients are bounded and $σ$ is non-degenerate. Furthermore, both $μ$ and $σ$ are assumed to be $C^{1}$ with intrinsic Lipschitz continuous derivative on $\mathbb{R}^{d}\setminus Θ$. We introduce, for the first time in literature, a Milstein-type method which can be used to approximate SDEs of this type for general $d \in \mathbb{N}$ and prove that this Milstein-type scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in terms of the number of steps. This method depends, in addition to evaluations of $W$ on a fixed grid, also on iterated integrals w.r.t. components of $W$, which can in general not be represented as functionals of $W$ evaluated at finitely many time points. We additionally prove that our suggested Milstein-type method is only dependent on evaluations of $W$ on a finite, fixed grid if $σ$ is additionally commutative. To obtain our main result we prove that a quasi-Milstein scheme achieves an $L_{p}$-error rate of order at least $3/4-$ in our setting if $μ$ is additionally continuous, which is of interest in itself. |
| title | Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient |
| topic | Probability |
| url | https://arxiv.org/abs/2505.15509 |