Skew-symmetric schemes for stochastic differential equations with non-Lipschitz drift: an unadjusted Barker algorithm

2024-05-23

arXiv.org

AI Summary

This work addresses time-homogeneous stochastic differential equations (SDEs) with non-Lipschitz drift coefficients, proposing an explicit numerical scheme for weak convergence simulation and ergodic distribution sampling—without requiring global Lipschitz assumptions. The method innovatively integrates skewed-symmetric increment sampling with the Barker acceptance mechanism, marking the first application of this combination to SDE discretization. Theoretical contributions include: (i) extending the Milstein–Tretyakov framework to accommodate non-Lipschitz drifts; (ii) rigorously establishing geometric convergence to the true invariant measure under fixed step size; and (iii) deriving quantitative pathwise error bounds and discrete distributional bias estimates. Numerical experiments confirm the algorithm’s high robustness and strong agreement with theoretical predictions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

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, 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.

Problem

• Develop skew-symmetric scheme for non-Lipschitz SDEs
• Analyze convergence and bias in ergodic diffusion simulation
• Extend weak convergence theory for non-Lipschitz drift SDEs

Innovation

• Skew-symmetric distribution sampling for SDEs
• Non-Lipschitz drift compatibility without global constraints
• Geometric convergence with fixed step-size bias analysis