On the performance of the Euler-Maruyama scheme for multidimensional SDEs with discontinuous drift coefficient

TL;DR

The paper addresses strong approximation of $d$-dimensional SDEs with discontinuous drift by proving that the Euler–Maruyama scheme achieves an $L_p$-error rate of at least $1/2-$ for all $d$ and all $p\ge 1$. The authors introduce a transformation $G$ that converts the SDE into a globally Lipschitz form, and develop a novel Itô formula for non-globally $C^2$ functions to control the transformed system. They combine moment bounds and occupation-time estimates for the Euler–Maruyama scheme with the transformation, obtaining sharp $L_p$-rates for both the time-continuous scheme and its piecewise linear interpolation. This extends known one-dimensional results to multidimensional settings with a hypersurface-structured discontinuity, enabling reliable numerical simulation of models with regime-switching drift in finance, insurance, and control.

Abstract

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 $Θ$.

Figures (10)

• Figure 1: Empirical $L_2$-error vs. number of time steps
• Figure 2: Empirical $L_2$-error vs. number of time steps: $a = -3, b = 1, x_{0} = (0,2)^\top$
• Figure 3: Empirical $L_1$-error vs. number of time steps: $a = -3, b = 1, x_{0} = (0,2)^\top$
• Figure 4: Histograms of realizations of $d_{1,n}$
• Figure 5: Histograms of realizations of $d_{2,n}$

Theorems & Definitions (99)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 1
• Remark 6
• Theorem 2
• Theorem 3
• Lemma 1