A Stable Numerical Scheme for Stochastic Differential Equations with Multiplicative Noise

We introduce a new approach for designing numerical schemes for stochastic differential equations (SDEs). The approach, which we have called the direction and norm decomposition method, proposes to approximate the required solution Xₜ by integrating the system of coupled SDEs that describes the evolution of the norm of Xₜ and its projection on the unit sphere. This allows us to develop an explicit scheme for stiff SDEs with multiplicative noise that shows a solid performance in various numerical experiments. Under general conditions, the new integrator preserves the almost sure stability of the solutions for any step-size, as well as the property of being distant from 0. The scheme also has a linear rate of weak convergence for a general class of SDEs with locally Lipschitz coefficients, and one-half strong order of convergence. (An erratum is attached.)