Abstract

We address the numerical solution of stochastic differential equations with multiplicative noise (SDEs) by means of balanced schemes. In particular, we study the design of balanced schemes that achieve the first order of weak convergence without involve the simulation of multiple stochastic integrals. We start by using the linear scalar SDE as a test problem to show that it is possible to construct almost sure stable first-order weak schemes based on the addition of stabilizing functions to the drift terms. Then, we consider multidimensional linear SDEs. In this case, we propose to find appropriate stabilizing weights through an optimization procedure. Finally, we illustrate the potential of the new methodology by solving the stochastic Duffing-Van Der Pol equation, which is a classical test non-linear SDE. Numerical experiments show a good performance of the numerical methods introduced in this paper.