Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm
<|reference_start|>Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm: 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, and also prove path-wise accuracy in a particular setting. 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.<|reference_end|>
@article{livingstone2024skew-symmetric,
    title={Skew-symmetric schemes for stochastic differential equations with
  non-Lipschitz drift: an unadjusted Barker algorithm},
    author={Samuel Livingstone, Nikolas N"usken, Giorgos Vasdekis and Rui-Yang
  Zhang},
    journal={arXiv preprint arXiv:2405.14373},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.14373},
    primaryClass={math.PR cs.NA math.NA stat.CO}
}