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SUMMARY:Skew-symmetric schemes for stochastic differential equations with 
 non-Lipschitz drift: an unadjusted Barker algorithm - Sam Livingstone (UCL
 )
DTSTART:20241101T140000Z
DTEND:20241101T150000Z
UID:TALK222349@talks.cam.ac.uk
CONTACT:Qingyuan Zhao
DESCRIPTION: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 numeric
 s\, 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. T
 he scheme is based on sampling increments at each time step from a skew-sy
 mmetric probability distribution\, with the level of skewness determined b
 y the drift and volatility of the underlying process. We show that as the 
 step-size decreases the scheme converges weakly to the diffusion of intere
 st\, and also show path-wise convergence for a model problem. We then cons
 ider the problem of simulating from the limiting distribution of an ergodi
 c diffusion process using the numerical scheme with a fixed step-size. We 
 establish conditions under which the numerical scheme converges to equilib
 rium at a geometric rate\, and quantify the bias between the equilibrium d
 istributions of the scheme and of the true diffusion process. Notably\, ou
 r results do not require a global Lipschitz assumption on the drift\, in c
 ontrast to those required for the Euler--Maruyama scheme for long-time sim
 ulation at fixed step-sizes. Our weak convergence result relies on an exte
 nsion of the theory of Milstein & Tretyakov to stochastic differential equ
 ations with non-Lipschitz drift.  This is joint work with Nikolas Nuesken\
 , Giorgos Vasdekis & Rui-yang Zhang.
LOCATION:Centre for Mathematical Sciences MR12\, CMS
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