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

TL;DR

This paper introduces a novel artificial barriers method for scalar SDEs, enabling boundary-preserving numerical schemes that handle non-globally Lipschitz coefficients while maintaining strong convergence rates.

Contribution

The paper proposes the artificial barriers approach for constructing boundary-preserving schemes for scalar SDEs, including those with non-globally Lipschitz coefficients, and introduces two new schemes, ABEM and ABEP.

Findings

Numerical experiments confirm the theoretical convergence rates.

The schemes effectively preserve boundary conditions in simulations.

Artificial barriers do not alter the original SDE solutions.

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.