BCAM Scientific Seminar | Skew-symmetric numerical schemes for stable long-time simulation of stochastic differential equations with non-Lipschitz drift

Abstract: 

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.  We also discuss comparisons to alternatives such as Tamed Euler, and preliminary results on strong convergence/multi-level extensions.  Based on joint work with Giorgos Vasdekis, Nikolas Nuesken & Ruiyang Zhang, Preprint available here: https://arxiv.org/abs/2405.14373.