Artificial Barriers for stochastic differential equations and for construction of boundary-preserving schemes

TL;DR

This work introduces artificial barriers to convert SDEs confined to a domain into nearby reflected problems, enabling explicit boundary-preserving schemes without relying on Lamperti-type transforms. By shifting the unattainable boundaries inward and applying RSDE discretizations (ABEM and ABEP), the authors establish strong convergence with rates matching RSDE theory, namely $\mathcal{O}(\Delta t^{1/2-\varepsilon})$ for ABEM and $\mathcal{O}(\Delta t^{1/2})$ for ABEP, while keeping error terms from barrier localization subdominant as $N\to\infty$. They provide rigorous $L^p$ convergence and almost-sure results, plus numerical experiments on geometric Brownian motion, Allen–Cahn type SDEs, and hat-function SDEs that confirm boundary-preservation and the predicted rates. The method offers a non-Lamperti, explicit framework for boundary-preserving discretization of SDEs with non-globally Lipschitz coefficients and restricted domains, with practical impact for simulations in physics, biology, and finance where domain integrity is essential.

Abstract

We develop the novel method of artificial barriers for scalar stochastic differential equations (SDEs) and use it to construct boundary-preserving numerical schemes for strong approximation of scalar SDEs, possibly with non-globally Lipschitz drift and diffusion coefficients, whose state-space is either bounded or half-bounded. The idea of artificial barriers is to augment the SDE with artificial barriers outside the state-space to not change the solution process, and then apply a boundary-preserving numerical scheme to the resulting reflected SDE (RSDE). This enables us to construct boundary-preserving numerical schemes that achieve the same strong convergence rate as the corresponding RSDE scheme. Based on the method of artificial barriers, we construct two boundary-preserving schemes that we call the Artificial Barriers Euler--Maruyama (ABEM) scheme and the Artificial Barriers Euler--Peano (ABEP) scheme, respectively. We provide numerical experiments for the ABEM scheme and the numerical results agree with the obtained theoretical results.

Figures (6)

• Figure 1: Path comparison of the Euler--Maruyama scheme (EM), the semi-implicit Euler--Maruyama scheme (SEM), the tamed Euler scheme (TE), and Artificial Barriers Euler--Maruyama scheme (ABEM) applied to the SDE for geometric Brownian motion in \ref{['eq:gBM']} with parameters $\lambda = 3$, $T = 1$, $x_{0}=0.1$, $d=1$ and $\Delta t = 0.02$.
• Figure 2: $L^{2}(\Omega)$-errors on the interval $[0,1]$ of the Artificial Barriers Euler--Maruyama scheme (ABEM) for the SDE for geometric Brownian motion in \ref{['eq:gBM']} for different choices of $\lambda>0$ and reference lines with slopes $1/2$ and $1/4$. Parameters: $T=1$, $x_{0}=0.1$, $d=1/2$ and $300$ Monte Carlo samples to approximate \ref{['eq:L2err']}.
• Figure 3: Path comparison of the Euler--Maruyama scheme (EM), the semi-implicit Euler--Maruyama scheme (SEM), the tamed Euler scheme (TE), and Artificial Barriers Euler--Maruyama scheme (ABEM) applied to the Allen-Cahn type SDE in \ref{['eq:ACsde']} with parameters $\lambda = 1$, $T = 1$, $x_{0}=0.9$, $d=1$ and $\Delta t = 0.02$.
• Figure 4: $L^{2}(\Omega)$-errors on the interval $[0,1]$ of the Artificial Barriers Euler--Maruyama scheme (ABEM) for the Allen--Cahn type SDE in \ref{['eq:ACsde']} for different choices of $\lambda>0$ and reference lines with slopes $1/2$ and $1/4$. Parameters: $T=1$, $x_{0}=0$, $d=1/2$ and $300$ Monte Carlo samples to approximate \ref{['eq:L2err']}.
• Figure 5: Path comparison of the EM, SEM, TE and ABEM schemes applied to the hat function SDE in \ref{['eq:hatSDE']} with parameters $\lambda = 3$, $T = 1$, $x_{0}=0.9$, $d=1$ and $\Delta t = 0.02$.

Theorems & Definitions (43)

• Remark 1
• Proposition 2
• Definition 3
• Lemma 4
• Definition 5
• Lemma 6
• Lemma 7
• proof : Proof of Lemma \ref{['lem:GNLipGB']}
• Definition 8
• Remark 9