We propose a new simple and explicit numerical scheme for time-homogeneous stochastic differential equations. The scheme is based on sampling increments at each time step from a skew-symmetric probability distribution, with the level of skewness determined by the drift and volatility of the underlying process. We show that as the step-size decreases the scheme converges weakly to the diffusion of interest. We then consider the problem of simulating from the limiting distribution of an ergodic diffusion process using the numerical scheme with a fixed step-size. We establish conditions under which the numerical scheme converges to equilibrium at a geometric rate, and quantify the bias between the equilibrium distributions of the scheme and of the true diffusion process. Notably, our results do not require a global Lipschitz assumption on the drift, in contrast to those required for the Euler--Maruyama scheme for long-time simulation at fixed step-sizes. Our weak convergence result relies on an extension of the theory of Milstein \& Tretyakov to stochastic differential equations with non-Lipschitz drift, which could also be of independent interest. We support our theoretical results with numerical simulations.

翻译：本文针对时齐随机微分方程提出了一种新颖、简单且显式的数值格式。该格式基于在每个时间步从斜对称概率分布中采样增量，其偏斜程度由底层过程的漂移项和波动率决定。我们证明了随着步长减小，该格式弱收敛于目标扩散过程。随后，我们考虑使用固定步长的数值格式模拟遍历扩散过程极限分布的问题。我们建立了数值格式以几何速率收敛到平衡态的条件，并量化了格式平衡分布与真实扩散过程平衡分布之间的偏差。值得注意的是，与Euler--Maruyama格式在固定步长下进行长时间模拟所需条件不同，我们的结果不要求漂移项满足全局Lipschitz条件。弱收敛结果的证明依赖于将Milstein与Tretyakov的理论推广至具有非Lipschitz漂移项的随机微分方程，该推广本身也具有独立的理论价值。我们通过数值模拟验证了理论结果。