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

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.