Corrected simulations for one-dimensional diffusion processes with naturally occurring boundaries.

Abstract

To simulate a diffusion process, a usual approach is to discretize the time in the associated stochastic differential equation. This is the approach used in the Euler method. In the present work we consider a one-dimensional diffusion process where the terms occurring, within the stochastic differential equation, prevent the process entering a region. The outcome is a naturally occurring boundary (which may be absorbing or reflecting). A complication occurs in a simulation of this situation. The term involving a random variable, within the discretized stochastic differential equation, may take a trajectory across the boundary into a "forbidden region." The naive way of dealing with this problem, which we refer to as the "standard" approach, is simply to reset the trajectory to the boundary, based on the argument that crossing the boundary actually signifies achieving the boundary. In this work we show, within the framework of the Euler method, that such resetting introduces a spurious force into the original diffusion process. This force may have a significant influence on trajectories that come close to a boundary. We propose a corrected numerical scheme, for simulating one-dimensional diffusion processes with naturally occurring boundaries. This involves correcting the standard approach, so that an exact property of the diffusion process is precisely respected. As a consequence, the proposed scheme does not introduce a spurious force into the dynamics. We present numerical test cases, based on exactly soluble one-dimensional problems with one or two boundaries, which suggest that, for a given value of the discrete time step, the proposed scheme leads to substantially more accurate results than the standard approach. Alternatively, the standard approach needs considerably more computation time to obtain a comparable level of accuracy to the proposed scheme, because the standard approach requires a significantly smaller time step.