  We present an explicit method for simulating stochastic differential
equations (SDEs) that have variable diffusion coefficients and satisfy the
detailed balance condition with respect to a known equilibrium density. In
Tupper and Yang (2012), we proposed a framework for such systems in which,
instead of a diffusion coefficient and a drift coefficient, a modeller
specifies a diffusion coefficient and a equilibrium density, and then assumes
detailed balance with respect to this equilibrium density. We proposed a
numerical method for such systems that works directly with the diffusion
coefficient and equilibrium density, rather than the drift coefficient, and
uses a Metropolis-Hastings rejection process to preserve the equilibrium
density exactly. Here we show that the method is weakly convergent with order
1/2 for such systems with smooth coefficients. We perform numerical experiments
demonstrating the convergence of the method for systems not covered by our
theorem, including systems with discontinuous diffusion coefficients and
equilibrium densities.

Numerical Analysis