Milstein-type methods for strong approximation of systems of SDEs with a discontinuous drift coefficient

TL;DR

This work addresses strong approximation of $d$-dimensional SDEs with discontinuous drifts by introducing a Milstein-type scheme suitable for general $d$. A key innovation is a transformation $G$ that maps the discontinuous-drift problem to a Lipschitz-coefficient problem, enabling rigorous $L_p$-error analysis and establishing a convergence rate of at least $3/4-\delta$ in the number of steps, with an enhanced rate when a perturbation surface $\Delta$ is absent. The scheme may require iterated Ito integrals, but under a commutativity condition on the diffusion, it reduces to evaluations of $W$ on a fixed grid, which is attractive for implementation. The paper also develops a quasi-Milstein variant achieving the same rate, provides moment and occupation-time estimates near the exceptional set, and supplies a constructive framework with concrete examples and numerical experiments that corroborate the theoretical findings. Overall, the results advance high-order strong approximation for SDEs with irregular drift by combining geometric transformation techniques with Milstein-type discretizations.

Abstract

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.

Figures (1)

• Figure 1: Empirical $L_2$-error vs. number of time steps: $a = -1, b = 1, x_{0} = (0,2)^\top$

Theorems & Definitions (62)

• Lemma 1
• proof
• Lemma 2
• Remark 1
• Theorem 1
• Theorem 2
• Proposition 1
• Proposition 2
• proof
• Lemma 3