Central limit theorem and Self-normalized Cramér-type moderate deviation for Euler-Maruyama Scheme

Abstract

We consider a stochastic differential equation and its Euler-Maruyama (EM) scheme, under some appropriate conditions, they both admit a unique invariant measure, denoted by $\pi$ and ${\pi }_{\eta }$ respectively ($\eta$ is the step size of the EM scheme). We construct an empirical measure ${\Pi }_{\eta }$ of the EM scheme as a statistic of ${\pi }_{\eta }$, and use Stein's method developed in \citet{FSX19} to prove a central limit theorem of ${\Pi }_{\eta }$. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting ${\eta }^{-1/2}({\Pi }_{\eta }(.)-\pi (.))$ into a martingale difference series sum $\mcl H_η$ and a negligible remainder $\mcl R_η$. We handle $\mcl H_η$ by the time-change technique for martingale, while prove that $\mcl R_η$ is exponentially negligible by concentration inequalities, which have their independent interest. Moreover, we show that SNCMD holds for $x=o({\eta }^{-1/6})$, which has the same order as that of the classical result in \citet{shao1999cramer,JSW03}.

To appear in Bernoulli, correct a small error in the proof of Lemma 4.1