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 X-t by integrating the system of coupled SDEs that describes the evolution of the norm of X-t 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.