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

I’ll present recent work involving skew-symmetric probability distributions, which have a long history of statistical applications and have enjoyed much recent interest. This work is mainly stochastic numerics, but was developed with statistical applications in mind and I will try to emphasise this during the talk. 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 show path-wise convergence for a model problem. 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. This is joint work with Nikolas Nuesken, Giorgos Vasdekis & Rui-yang Zhang.

• Speaker: Sam Livingstone (UCL)
• Friday 01 November 2024, 14:00–15:00
• Venue: Centre for Mathematical Sciences MR12, CMS.
• Series: Statistics; organiser: Qingyuan Zhao.