Skew-symmetric Numerical Schemes for Stochastic Differential Equations with Non-Lipschitz Drift: Weak Convergence and Long-Time Behavior

The authors propose a new class of explicit numerical schemes for simulating stochastic differential equations with non-Lipschitz drift. The schemes are 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. The authors establish weak convergence of the schemes to the true diffusion process as the step-size decreases, as well as geometric ergodicity and bias control for the long-time behavior of the numerical schemes.