A nonstandard Euler-Maruyama scheme
For researchers in stochastic numerics, this provides a new scheme with better domain invariance properties, though it is an incremental improvement over existing methods.
The paper introduces a nonstandard Euler-Maruyama scheme for SDEs, proving strong convergence under local Lipschitz and linear growth conditions, and demonstrating domain invariance preservation. Numerical tests on geometric Brownian motion show improved behavior over standard Euler-Maruyama and balanced implicit methods.
We construct a nonstandard finite difference numerical scheme to approximate stochastic differential equations (SDEs) using the idea of weighed step introduced by R.E. Mickens. We prove the strong convergence of our scheme under locally Lipschitz conditions of a SDE and linear growth condition. We prove the preservation of domain invariance by our scheme under a minimal condition depending on a discretization parameter and unconditionally for the expectation of the approximate solution. The results are illustrated through the geometric Brownian motion. The new scheme shows a greater behavior compared to the Euler-Maruyama scheme and balanced implicit methods which are widely used in the literature and applications.