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

2020/12/08 by Jianya Lu, Yuzhen Tan, Lu, Jianya +3 · 5 citations
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #Markov Chains and Monte Carlo Methods

paper · pdf · doi:10.48550/arxiv.2012.04328

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 π and π_η respectively (η is the step size of the EM scheme). We construct an empirical measure Π_η of the EM scheme as a statistic of π_η, and use Stein's method developed in \citetFSX19 to prove a central limit theorem of Π_η. The proof of the self-normalized Cramér-type moderate deviation (SNCMD) is based on a standard decomposition on Markov chain, splitting η^-1/2(Π_η(.)-π(.)) 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(η^-1/6), which has the same order as that of the classical result in \citetshao1999cramer,JSW03.

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