Modelling and numerical methods for state-dependent diffusions

When modelling diffusive systems with stochastic differential equations, a question about interpretations of the stochastic integral often arises. Using simulations of a random Lorentz gas model, we show that given only the diffusion coefficient, for a diffusive system without external force, the system is underdetermined. By varying one free parameter, the prediction from different interpretations can hold true. However, for a diffusive system satisfying detailed balance condition, we show that it is uniquely determined by the equilibrium distribution in addition to the diffusion coefficient. We propose an explicit method for simulating stochastic differential equations in this formulation. Our numerical scheme introduces Metropolis-Hastings step-rejections to preserve the exact equilibrium distribution and works directly with the diffusion coefficient rather than the drift coefficient. We show that the numerical scheme 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 coefficients.