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.

中文翻译：

---

随机微分方程的一阶弱平衡方案

我们通过平衡方案来解决带有乘性噪声（SDE）的随机微分方程的数值解。特别是，我们研究了平衡方案的设计，该方案实现了弱收敛的一阶而不涉及多个随机积分的模拟。我们从使用线性标量SDE作为测试问题开始，以表明可以通过在漂移项上添加稳定函数来构造几乎确定的稳定一阶弱方案。然后，我们考虑多维线性SDE。在这种情况下，我们建议通过优化程序找到合适的稳定权重。最后，我们通过求解随机Duffing-Van Der Pol方程来说明这种新方法的潜力，Duffing-Van Der Pol方程是经典的测试非线性SDE。